Coulomb gas integrals for commuting SLEs:
Schramm’s formula and Green’s function
arXiv:1701.03698v1 [math.PR] 13 Jan 2017
Jonatan Lenells and Fredrik Viklund
KTH Royal Institute of Technology
January 16, 2017
Abstract
We use methods developed in conformal field theory to produce martingale observables for systems of commuting multiple SLE curves. In the case of two curves
started from distinct points and growing towards infinity, we use these observables
to determine rigorously and explicitly the Green’s function and Schramm’s formula.
As corollaries, we obtain proofs of “fusion” formulas, some of which have been predicted in the physics literature. Our approach does not need a priori information
on the regularity of the SLE observables, but does require a detailed analysis of the
regularity and asymptotics of certain Coulomb gas contour integrals. These integrals
are natural generalizations of the classic hypergeometric functions and are interesting
in their own right. We indicate a method for computing the relevant asymptotics of
these integrals to all orders.
Contents
1 Introduction
1.1 Outline of the paper . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.2 Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3
4
5
2 Main results
2.1 Schramm’s formula . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.2 Green’s function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.3 Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5
6
8
12
3 Preliminaries
3.1 Schramm-Loewner evolution . . . . . . . . . . . . . . . .
3.1.1 SLEκ (ρ) . . . . . . . . . . . . . . . . . . . . . . .
3.1.2 Relationship between multiple SLE and SLEκ (ρ)
3.1.3 Two-sided radial SLE and radial parametrization
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4 Martingale observables as CFT correlation
4.1 Screening . . . . . . . . . . . . . . . . . . .
4.2 Prediction of Schramm’s formula . . . . . .
4.3 Prediction of the Green’s function . . . . .
functions
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5 Schramm’s formula
5.1 Proof of Theorem 2.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.2 The special case κ = 4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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6 Green’s function
6.1 Existence of the Green’s function: Proof of Proposition 2.12 . . . . . . . .
6.2 Probabilistic representation for GCFT : Proof of Proposition 2.13 . . . . . .
6.2.1 The function h(θ1 , θ2 ) . . . . . . . . . . . . . . . . . . . . . . . . .
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7 Two paths getting near the same point: Proof of Lemma 2.9
42
8 Fusion
8.1 Schramm’s formula . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
8.2 Green’s function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
47
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9 The function G(z, ξ 1 , ξ 2 ) when α is an integer
9.1 A representation for h . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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54
10 Asymptotics of Coulomb gas integrals
10.1 Screening integrals . . . . . . . . . . . . .
10.2 The hypergeometric case of N = 3 . . . .
10.3 Asymptotics of F as w2 → 1 . . . . . . . .
10.4 Asymptotics of F as w1 → 0 . . . . . . . .
10.5 Asymptotics of F as w1 → 0 and w2 → 0 .
10.6 Asymptotics of F as w1 → 0 and w2 → 1 .
10.7 Some basic estimates . . . . . . . . . . . .
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11 Differential equations
75
A Estimates for Schramm’s formula
A.1 Properties of the function J(z, ξ) . . . . . . . . . . . . . . . . . . . . . . .
A.2 Existence and regularity of P . . . . . . . . . . . . . . . . . . . . . . . . .
76
76
85
B Estimates for Green’s function
B.1 The asymptotic sectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
B.2 Representations for h . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
B.3 Proof of Lemma 6.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
92
92
93
97
2
1
Introduction
Schramm-Loewner evolution (SLE) processes are universal lattice size scaling limits of
interfaces in critical planar lattice models. By the Markovian property of SLE, geometric observables determine martingales with respect to the natural SLE filtration. Such
martingale observables satisfy differential equations, which can sometimes be used to find
explicit formulas for the observables, or be used as a basis for estimates, see for instance
[35, 9]. A related problem is that of constructing martingales carrying some specific
geometric information about the SLE process.
The differential equations are usually derived using Itô calculus, assuming sufficient
regularity to apply Itô’s formula. If a solution with suitable boundary values can be found,
one way to proceed is to perform a probabilistic argument using the solution’s boundary
behavior to show that it actually represents the desired quantity. In the simplest cases
the differential equation is an ODE, but generically it is a semi-elliptic PDE in several
variables and it is difficult to find solutions with desired boundary data. (But see e.g.
[15, 25] for cases when solutions are available.) Depending on the amount of available
information, non-trivial regularity issues may need to be resolved when analyzing the
PDEs, see, e.g, [16]. Seeking ways to construct solutions and methods for extracting
information from them therefore seems worthwhile.
The basic PDEs that arise in SLE theory also arise in conformal field theory (CFT), see
e.g. [8, 10, 5, 16]. On the other hand, conformal field theory provides ideas and methods for
how to systematically construct solution candidates, see e.g. [6, 7, 23]. We will make use
of the Coulomb gas framework of [23] which models the CFTs using certain Gaussian free
field correlation functions. Very briefly, CFT correlations involving special fields inserted
on the boundary give rise to SLE martingales and thereby PDE solutions. By making
additional, carefully chosen, field insertions, the scaling behavior at the insertion points
can in some cases be prescribed. In this way, using a “calculus of scaling dimensions”,
many SLE martingale observables were recovered in [23] as CFT correlation functions.
The purpose of this paper is to construct one-point martingale observables (equivalently, to solve the corresponding PDEs) related to specific geometric information for
multiple commuting SLEs by exploiting ideas from CFT. In the process we will suggest
an approach for rigorously deriving at least some natural observables in this setting. The
argument proceeds through three steps:
(1) The first step generates a non-rigorous prediction for the observable by using a screening argument based on ideas from CFT [13, 23, 3]. (See also [12, 25] and the references
in the latter.) The prediction is expressed as a contour integral with an explicit integrand. We call these integrals screening integrals or Coulomb gas integrals, see Section
4. The main difficulty is to choose the appropriate integration contour, but this choice
can be simplified by considering appropriate limits.
(2) The second step is to prove that the prediction from Step 1 satisfies the correct boundary conditions. This technical step involves the computation of somewhat complicated
integral asymptotics, but we indicate an approach for computing such asymptotics in
Section 10.
3
(3) The last step is to use probabilistic methods together with the estimates of Step 2 to
rigorously establish that the prediction from Step 1 gives the correct quantity.
We analyze two examples in detail. Both examples involve two curves aiming for ∞
with one marked point in the interior, but we will indicate how the arguments can be
generalized to more complicated configurations.
The first example concerns the probability that the system of SLEs passes to the right
of a given interior point; that is, the analog of Schramm’s formula [35]. This probability
obviously depends only on the behavior of the leftmost curve. (So it is really an SLEκ (2)
observable.) The main difficulty in this case lies in implementing Steps 1 and 2.
The second example concerns the limiting renormalized probability that the system
of SLEs passes near a given point, that is, the Green’s function. We first give a proof of
the non-trivial fact that the commuting Green’s function actually exists as a limit. The
main step is to verify existence in the case when only one of the two curves grows. (More
precisely, we prove the existence of the SLEκ (ρ) Green’s function, where ρ is in an interval
and the force point is on the boundary.) The proof gives a representation formula in terms
of an expectation for two-sided radial SLE stopped at its target point; this is similar to
the main result of [2]. In Step 1, the prediction is made by taking a linear combination
of screening integrals to cancel a leading order term which has the wrong asymptotics
(thereby matching the asymptotics we expect). Then, in Step 2, we carefully analyze the
candidate solution and estimate its boundary behavior. Lastly, given the estimates from
Step 2, we show that the candidate observable enjoys the same probabilistic representation
as the Green’s function defined as a limit – so they must agree.
By letting the seed points of the SLEs collapse, we obtain rigorous proofs of fusion formulas as corollaries. One can verify a posteriori that these limiting one-point observables
satisfy specific third-order ODEs that can also be obtained from the so-called fusion rules
of conformal field theory, see e.g. [12]. In fact, in the case of the Schramm probability,
the formulas we prove here were predicted using fusion in [20]. The formulas for the fused
Green’s functions appear to be new. The interpretation of fusion in SLE theory as successive merging of seeds, and the highly non-trivial fact that fused one-point observables
satisfy higher order ODEs, was rigorously established in [16]. The ODEs for the Schramm
formula for several fused paths were derived rigorously in [16] and the two-path formula
in the special case κ = 8/3 (also allowing for two interior points) was established in [4].
As can be gathered from the length of the paper, there are many details to handle.
In particular, the asymptotic analysis of the screening integrals is quite involved. These
integrals are natural generalizations of hypergeometric functions and are interesting in
their own right. Similar generalized hypergeometric functions have been considered before
in related contexts, see e.g. [13, 15, 25, 26, 22]. However, we have not been able to find
the required analytic estimates in the litterature. In Section 10, we propose a method for
establishing the asymptotics of these integrals to all orders in a rather general setting.
1.1
Outline of the paper
The main results of the paper are stated in Section 2, while we review some aspects of
multiple SLEκ and SLEκ (ρ) processes in Section 3.
4
In Section 4, we review and use ideas from conformal field theory to generate predictions for Schramm’s formula and Green’s function for commuting SLE with two curves
growing toward infinity as Coulomb gas integrals.
In Section 5, we prove rigorously that the predicted Schramm’s formula indeed gives
the probability that a given point lies to the left of both curves. The proof relies on
a number of technical asymptotic estimates; proofs of these estimates are collected in
Appendix A.
In Section 6, we give a rigorous proof that the predicted Green’s function equals the
renormalized probability that the system passes near a given point. The proof relies both
on pure SLE estimates (established in Section 6) and on asymptotic estimates for contour
integrals (established in Section 10 and Appendix B).
In Section 7, we prove a lemma which expresses the fact that it is very unlikely that
both curves in a commuting system get near a given point.
In Section 8, we consider the special case of two fused curves, i.e., the case when
both curves in the commuting system start at the same point. In the case of Schramm’s
formula, this provides rigorous proofs of some predictions due to Gamsa and Cardy for
Schramm’s formula presented in [20].
In Section 9, we derive formulas for the Green’s function in the special case when 8/κ
is an integer.
In Section 10, we suggest an approach for computing certain asymptotics to all orders
for a class of contour integrals which generalize the classic hypergeometric functions. The
proofs in Appendix B are an application of this approach to the contour integral relevant
for the Green’s function.
In Section 11, we discuss our results from the point of view of differential equations.
1.2
Acknowledgements
Lenells acknowledges support from the European Research Council, Consolidator Grant
No. 682537, the Swedish Research Council, Grant No. 2015-05430, and the Gustafsson
Foundation, Sweden. Viklund acknowledges support from the Knut and Alice Wallenberg
Foundation, the Swedish Research Council, the National Science Foundation, and the
Gustafsson Foundation, Sweden.
It is our pleasure to thank Julien Dubédat and Nam-Gyu Kang for interesting and
useful discussions and Tom Alberts, Nam-Gyu Kang, and Nikolai Makarov for sharing
with us ideas from their preprint [3].
2
Main results
Before stating the main results, we review the definition of a system of two commuting
SLE paths {γj }21 in the upper half-plane H := {Im z > 0} growing toward infinity.
Let 0 < κ < 8. Let (ξ 1 , ξ 2 ) ∈ R2 with ξ 1 6= ξ 2 . The Loewner equation corresponding
to two growing curves is
dgt (z) =
λ1 (t)dt
λ2 (t)dt
+
,
1
gt (z) − ξt
gt (z) − ξt2
5
g0 (z) = z,
(2.1)
where ξt1 and ξt2 , t > 0, are the driving terms for the two curves and the growth speeds
λj (t) satisfy λj (t) > 0. The solution of (2.1) is a family of conformal maps (gt (z))t>0
called the Loewner chain of (ξt1 , ξt2 ). The system of two commuting SLEs started from
(ξ 1 , ξ 2 ) is obtained by taking ξt1 and ξt2 as solutions of the system
q

κ
2 (t)
dξt1 = λ1 (t)+λ
dt
+
λ1 (t)dBt1 ,
1
2
ξt −ξt
q2
2 (t)
dξ 2 = λ1 (t)+λ
dt + κ λ2 (t)dB 2 ,
2
1
t
t
2
ξt −ξt
ξ01 = ξ 1 ,
ξ02 = ξ 2 ,
(2.2)
where Bt1 and Bt2 are independent standard Brownian motions with respect to some
measure P = Pξ1 ,ξ2 . The paths are defined by
γj (t) = lim gt−1 (ξtj + iy),
y↓0
γj,t := γj [0, t],
j = 1, 2.
(2.3)
For j = 1, 2, γj,∞ is a continuous curve connecting ξ j with ∞ in H. Given z ∈ C \ (γ1,∞ ∪
γ2,∞ ), we let Υ∞ (z) denote 1/2 times the conformal radius of H \ (γ1,∞ ∪ γ2,∞ ) seen from
z. It can be shown that the system (2.1) is commuting in the sense that the order in
which the two curves are grown does not matter [15]. Since our theorems only concern
the distribution of the fully grown curves γ1,∞ and γ2,∞ , the choice of growth speeds is
irrelevant.
Let us also recall the definition of (chordal) SLEκ (ρ) for a single path γ1 in H growing
toward infinity. Let ρ ∈ R and let (ξ 1 , ξ 2 ) ∈ R2 with ξ 1 6= ξ 2 . Let Wt be a standard
Brownian motion with respect to some measure Pρ . Then SLEκ (ρ) started from (ξ 1 , ξ 2 )
is defined by the equations
2/κ
, g0 (z) = z,
gt (z) − ξt1
ρ/κ
dξt1 = 1
dt + dWt , ξ01 = ξ 1 .
ξt − gt (ξ 2 )
∂t gt (z) =
When referring to SLEκ (ρ) started from (ξ 1 , ξ 2 ), we always assume that the curve starts
from the first point of the tuple (ξ 1 , ξ 2 ) while the second point (in this case ξ 2 ) is the force
point. The SLEκ (ρ) path γ1 (t) is defined as in (2.3). Given z ∈ C \ γ1,∞ , we let Υ∞ (z)
denote 1/2 times the conformal radius of H \ γ1,∞ seen from z.
2.1
Schramm’s formula
Our first result provides an explicit expression for the probability that an SLEκ (2) path
passes to the right of a given point. The probability is expressed in terms of the function
M(z, ξ) defined for z ∈ H and ξ > 0 by
α
α
α
α
M(z, ξ) = y α−2 z − 2 (z − ξ)− 2 z̄ 1− 2 (z̄ − ξ)1− 2
×
Z z
α
α
(u − z)α (u − z̄)α−2 u− 2 (u − ξ)− 2 du,
(2.4)
z̄
where α = 8/κ > 1 and the integration contour from z̄ to z passes to the right of ξ, see
Figure 1.
6
Im z
z
0
Re z
ξ
z̄
Figure 1. The integration contour used in the definition (2.4) of M(z, ξ) is a path from
z̄ to z which passes to the right of ξ.
Theorem 2.1 (Schramm’s formula for SLEκ (2)). Let 0 < κ < 8. Let ξ > 0 and consider
chordal SLEκ (2) started from (0, ξ). Then the probability P (z, ξ) that a given point z =
x + iy ∈ H lies to the left of the curve is given by
1
P (z, ξ) =
cα
Z ∞
Re M(x0 + iy, ξ)dx0 ,
x
z ∈ H,
ξ > 0,
(2.5)
where the normalization constant cα ∈ R is defined by
2π 3/2 Γ
cα = −
Γ
α−1
Γ 3α
2
2
α 2
Γ(α)
2
−1
.
(2.6)
The proof of Theorem 2.1 will be given in Section 5. The form of the definition (2.5)
of P (z, ξ) is motivated by the CFT and screening considerations of Section 4.
A point z ∈ H lies to the left of both curves in a commuting system iff it lies to
the left of the leftmost curve. Since each of the two curves of a commuting process has
the distribution of an SLEκ (2) (see Section 3.1.2), Theorem 2.1 immediately implies the
following result for commuting SLE.
Corollary 2.2 (Schramm’s formula for two commuting SLEs). Let 0 < κ < 8. Let ξ > 0
and consider a system of two commuting SLEκ curves in H started from (0, ξ) and growing
toward infinity. Then the probability P (z, ξ) that a given point z = x + iy ∈ H lies to the
left of both curves is given by (2.5).
Corollary 2.2 together with translation invariance immediately yields an expression
for the probability that a point z lies to the left of a system of two SLEs started from
two arbitrary points (ξ1 , ξ2 ) in R. The probabilities that z lies to the right of or between
7
the two curves then follow by symmetry. For completeness, we formulate this as another
corollary.
Corollary 2.3. Let 0 < κ < 8. Suppose −∞ < ξ 1 < ξ 2 < ∞ and consider a system of
two commuting SLEκ curves in H started from (ξ 1 , ξ 2 ) and growing toward infinity. Let
P (z, ξ) denote the function in (2.5). Then the probability Plef t (z, ξ 1 , ξ 2 ) that a given point
z = x + iy ∈ H lies to the left of both curves is given by
Plef t (z, ξ 1 , ξ 2 ) = P (z − ξ 1 , ξ 2 − ξ 1 );
the probability Pright (z, ξ 1 , ξ 2 ) that a point z ∈ H lies to the right of both curves is
Pright (z, ξ 1 , ξ 2 ) = P (−z̄ + ξ 2 , ξ 2 − ξ 1 );
and the probability Pmiddle (z, ξ 1 , ξ 2 ) that z lies between the two curves is given by
Pmiddle (z, ξ 1 , ξ 2 ) = 1 − Plef t (z, ξ 1 , ξ 2 ) − Pright (z, ξ 1 , ξ 2 ).
By letting ξ → 0+, we obtain proofs of formulas for “fused” paths. This gives a proof
of the predictions of [20] where these formulas were derived by solving a third order ODE
obtained from so-called fusion rules. We note that even given the explicit predictions of
[20], it is not clear how to proceed to verify them rigorously. Indeed, as soon the evolution
starts, the tips of the curves are separated and the system leaves the fused state. However,
[16] provides a different rigorous approach by exploiting the hypoellipticity of the PDEs
to show that the fused observables satisfy the higher order ODEs. The formula in the
special case κ = 8/3 was proved in [4] using Cardy and Simmons’ prediction [11] for a
two-point Schramm formula. See Theorem 8.1 for a complete statement and details.
Corollary 2.4. Let 0 < κ < 8 and define Pf usion (z) = limξ↓0 P (z, ξ), where P (z, ξ) is as
in (2.5). Then
Γ( α )Γ(α)
Pf usion (z) = 2−α 2 3α
2
πΓ( 2 − 1)
Z ∞
x
y
S(t0 )dt0 ,
where the real-valued function S(t) is defined by
1 α
α 1
+ , 1 − , ; −t2
2
2
2 2
α
α
2Γ(1 + 2 )Γ( 2 )t
α 3 α 3
2
− 1 α
− , ; −t
,
2 F1 1 + ,
2 2
2 2
Γ( 2 + 2 )Γ(− 21 + α2 )
S(t) = (1 + t2 )1−α
2.2
2 F1
t ∈ R.
Green’s function
Our second main result provides an explicit expression for the Green’s function for SLEκ (2).
Let α = 8/κ. Define the function I(z, ξ 1 , ξ 2 ) for z ∈ H and −∞ < ξ 1 < ξ 2 < ∞ by
I(z, ξ 1 , ξ 2 ) =
Z (z+,ξ 2 +,z−,ξ 2 −)
α
α
(u − z)α−1 (u − z̄)α−1 (u − ξ 1 )− 2 (ξ 2 − u)− 2 du,
A
8
(2.7)
z
ξ2
ξ1
Re z
A
z̄
Figure 2. The Pochhammer integration contour in (2.7) is the composition of four loops
based at the point A > ξ2 .
where A > ξ 2 is a basepoint and the Pochhammer integration contour is displayed in
Figure 2. More precisely, the integration contour begins at the base point A, encircles the
point z once in the positive sense, returns to A, encircles ξ 2 once in the positive sense,
returns to A, and so on. The points z̄ and ξ 1 are exterior to all loops. The factors in
the integrand take their principal values at the starting point and are then analytically
continued along the contour.
For α ∈ (1, ∞) \ Z, we define the function G(z, ξ 1 , ξ 2 ) by
1
1
G(z, ξ 1 , ξ 2 ) = y α+ α −2 |z − ξ 1 |1−α |z − ξ 2 |1−α Im e−iπα I(z, ξ 1 , ξ 2 ) ,
ĉ
z ∈ H,
ξ1 < ξ2,
(2.8)
where the constant ĉ = ĉ(κ) is given by
4 sin2
ĉ =
πα
2
sin(πα)Γ 1 −
α
2
Γ(α)
Γ
3α
2
−1
with α =
8
.
κ
(2.9)
We extend this definition of G(z, ξ 1 , ξ 2 ) to all α > 1 by continuity. The following lemma
shows that even though ĉ vanishes as α approaches an integer, the function G(z, ξ 1 , ξ 2 )
has a continuous extension to integer values of α.
Lemma 2.5. For each z ∈ H and each (ξ 1 , ξ 2 ) ∈ R2 with ξ 1 < ξ 2 , G(z, ξ 1 , ξ 2 ) can be
extended to a continuous function of α ∈ (1, ∞).
Proof. See Section 9.
The CFT and screening considerations described in Section 4 suggest that G is the
Green’s function for SLEκ (2) started from (ξ 1 , ξ 2 ); that is, that G(z, ξ 1 , ξ 2 ) provides the
normalized probability that an SLEκ (2) path originating from ξ 1 with force point ξ 2
passes through an infinitesimal neighborhood of z. The following theorem establishes this
rigorously.
9
Theorem 2.6 (Green’s function for SLEκ (2)). Let 0 < κ 6 4. Let −∞ < ξ 1 < ξ 2 < ∞
and consider chordal SLEκ (2) started from (ξ 1 , ξ 2 ). Then, for each z = x + iy ∈ H,
lim d−2 P2ξ1 ,ξ2 (Υ∞ (z) 6 ) = c∗ G(z, ξ 1 , ξ 2 ),
(2.10)
→0
where P2 is the SLEκ (2) measure, the function G is defined in (2.8), and the constant
c∗ = c∗ (κ) is defined by
c∗ = R π
0
2
2Γ (1 + 2a)
=√ sin4a xdx
πΓ 21 + 2a
with
a=
2
.
κ
(2.11)
The proof of Theorem 2.6 will be presented in Section 6.
Remark 2.7. The function G(z, ξ 1 , ξ 2 ) can be written as
G(z, ξ 1 , ξ 2 ) = (Im z)d−2 h(θ1 , θ2 ),
z ∈ H, −∞ < ξ 1 < ξ 2 < ∞,
(2.12)
where h is a function of θ1 = arg(z − ξ 1 ) and θ2 = arg(z − ξ 2 ). This is consistent with the
expected translation invariance and scale covariance of the Green’s function.
Remark 2.8. Formulas for G(z, ξ 1 , ξ 2 ) when α is an integer are derived in Section 9.
For α = 2, 3, 4 (corresponding to κ = 4, 8/3, 2), the function h(θ1 , θ2 ) in (2.12) is given
explicitly in (9.18), (9.19), and (9.20), respectively.
It is possible to derive an explicit expression for the Green’s function for a system of
two commuting SLEs as a consequence of Theorem 2.6. To this end, we need a correlation
estimate which expresses the fact that it is very unlikely that both curves in a commuting
system pass near a given point z ∈ H.
Lemma 2.9. Let 0 < κ 6 4. Then,
i
h
lim d−2 Pξ1 ,ξ2 (Υ∞ (z) 6 ) = lim d−2 P2ξ1 ,ξ2 (Υ∞ (z) 6 ) + P2ξ2 ,ξ1 (Υ∞ (z) 6 ) ,
↓0
↓0
where Pξ1 ,ξ2 denotes the law of a system of two commuting SLEκ in H started from
(ξ 1 , ξ 2 ) and aiming for ∞, and P2ξ1 ,ξ2 denotes the law of chordal SLEκ (2) in H started
from (ξ 1 , ξ 2 ).
The proof of Lemma 2.9 will be given in Section 7.
Assuming Lemma 2.9, it follows immediately from Theorem 2.6 that the Green’s
function for a system of commuting SLEs started from (−ξ, ξ) is given by
Gξ (z) = G(z, −ξ, ξ) + G(−z̄, −ξ, ξ),
z ∈ H,
ξ > 0.
In other words, given a system of two commuting SLEκ paths started from −ξ and ξ
respectively, Gξ (z) provides the normalized probability that at least one of the two curves
passes through an infinitesimal neighborhood of z. We formulate this as a corollary.
10
Corollary 2.10 (Green’s function for two commuting SLEs). Let 0 < κ 6 4. Let ξ > 0
and consider a system of two commuting SLEκ paths in H started from (−ξ, ξ) and growing
towards ∞. Then, for each z = x + iy ∈ H,
lim d−2 P−ξ,ξ (Υ∞ (z) 6 ) = c∗ Gξ (z),
(2.13)
→0
where d = 1 + κ/8, the constant c∗ = c∗ (κ) is given by (2.11), and the function Gξ is
defined for z ∈ H and ξ > 0 by
1
1
Gξ (z) = y α+ α −2 |z + ξ|1−α |z − ξ|1−α Im e−iπα (I(z, −ξ, ξ) + I(−z̄, −ξ, ξ)) .
ĉ
Remark 2.11. If the commuting system is started from two arbitrary points (ξ 1 , ξ 2 ) ∈ R
with ξ 1 < ξ 2 , then it follows immediately from (2.13) and translation invariance that
d−2
lim →0
Pξ1 ,ξ2 (Υ∞ 6 ) = c∗ G ξ2 −ξ1
2
ξ1 + ξ2
z−
.
2
The proof of Theorem 2.6 will consist of establishing two independent propositions,
which when combined imply Theorem 2.6. The first of these propositions (Proposition
2.12) establishes existence of a Green’s function for SLEκ (ρ) and provides a representation
for this Green’s function in terms of an expectation with respect to two-sided radial SLEκ .
For the proof of Theorem 2.6, we only need this proposition for ρ = 2. However, since it
is no more difficult to state and prove it for a suitable range of positive ρ, we consider the
general case.
Proposition 2.12 (Existence and representation of Green’s function for SLEκ (ρ)). Let
0 < κ 6 4 and 0 6 ρ < 8 − κ. Given two points ξ 1 , ξ 2 ∈ R with ξ 1 < ξ 2 , consider chordal
SLEκ (ρ) started from (ξ 1 , ξ 2 ). Then, for each z ∈ H,
lim d−2 Pρξ1 ,ξ2 (Υ∞ 6 ) = c∗ Gρ (z, ξ 1 , ξ 2 ),
↓0
where the SLEκ (ρ) Green’s function Gρ is given by
h
(ρ)
Gρ (z, ξ 1 , ξ 2 ) = G(z − ξ 1 ) E∗ξ1 ,z MT
i
.
(2.14)
Here G(z) = (Im z)d−2 sin4a−1 (arg z) is the Green’s function for chordal SLEκ in H from
(ρ)
0 to ∞, the martingale Mt is defined in (3.7), E∗ξ1 ,z denotes expectation with respect
to two-sided radial SLEκ from ξ 1 through z, stopped at T , the hitting time of z, and the
constant c∗ is given by (2.11).
The next result (Proposition 2.13) shows that the function G(z, ξ 1 , ξ 2 ) predicted by
CFT and defined in (2.8) can be represented in terms of an expectation with respect
to two-sided radial SLEκ . Since this representation coincides with the representation in
(2.14), Theorem 2.6 will follow immediately once we establish Propositions 2.12 and 2.13.
11
Proposition 2.13 (Representation of G). Let 0 < κ 6 4 and let ξ 1 , ξ 2 ∈ R with ξ 1 < ξ 2 .
The function G(z, ξ 1 , ξ 2 ) defined in (2.8) satisfies
h
(2)
G(z, ξ 1 , ξ 2 ) = G(z − ξ 1 ) E∗ξ1 ,z MT
i
,
z ∈ H,
0 < ξ < ∞,
(2.15)
where G(z) = (Im z)d−2 sin4a−1 (arg z) is the Green’s function for chordal SLEκ in H from
0 to ∞ and E∗ξ1 ,z denotes expectation with respect to two-sided radial SLEκ from ξ 1 through
z, stopped at T , the hitting time of z.
Remark 2.14. Note that equation (2.15) gives a formula for the two-sided radial SLE
observable,
h
i
G(z, ξ 1 , ξ 2 )
(2)
E∗ξ1 ,z MT =
G(z − ξ 1 )
and as a consequence we obtain smoothness and the fact that it satisfies the expected
PDE.
The proofs of Propositions 2.12 and 2.13 are presented in Sections 6.1 and 6.2, respectively.
In Section 8.2, we obtain fusion formulas by letting ξ → 0+. The formulas simplify
for some values of κ.
Proposition 2.15 (Fused Green’s functions). Suppose κ = 4, 8/3, or 2. Consider a
system of two fused commuting SLEκ paths in H started from 0 and growing toward ∞.
Then, for each z = x + iy ∈ H,
lim d−2 P0,0+ (Υ∞ (z) 6 ) = c∗ (Gf (z) + Gf (−z̄)),
→0
where d = 1 + κ/8, the constant c∗ = c∗ (κ) is given by (2.11), and the function Gf is
defined by
Gf (x + iy) = y d−2 hf (θ)
with hf (θ) given explicitly by
hf (θ) =
2.3

2


 π (sin θ − θ cos θ) sin θ,
8
2
15π (4θ − 3 sin 2θ + 2θ cos 2θ) sin θ,


 1 (27 sin θ + 11 sin 3θ − 6θ(9 cos θ +
12π
cos 3θ)) sin3 θ,
κ = 4,
κ = 8/3,
κ = 2,
0 < θ < π.
Remarks
We end this section by making a few remarks.
• We believe the method used in this paper will generalize to produce analogous results
for observables for N > 3 commuting SLE paths depending on one interior point.
This would require N − 1 screening insertions, and the integrals will then be N − 1
iterated contour integrals.
12
• In [25, 26] screening integrals for SLE boundary observables (such as the ordered
multipoint boundary Green’s function) are given and shown to be closely related
to a particular quantum group. In fact, this algebraic structure is used to systematically make the difficult choices of integration contours. It seems reasonable to
expect that a similar connection exists in our setting as well, allowing for an efficient
generalization to several commuting SLE curves, but we will not pursue this here.
• Another way of viewing the system of two commuting SLEs growing towards ∞ is as
one SLE path conditioned to hit a boundary point, also known as two-sided chordal
SLE. Indeed, the extra ρ = 2 at the second seed point forces a ρ = κ − 8 at ∞.
• A comparison with the representation of Appell’s F1 function as an Euler integral
(see e.g. [17]) shows that our screening integrals in the simplest case can be viewed
as analytic continuations of Appell’s function. It is possible that our asymptotic
estimates can be derived from known facts about these functions, but we have not
been able to find the needed bounds in the litterature. Besides, our approach generalizes to accomodate for multiple screening insertions, corresponding, for example,
to several commuting SLEs. We refer to Section 10 for additional discussion.
• Suppose one has an SLEκ martingale and wants to construct a similar martingale
for SLEκ (ρ). The first idea that comes to mind is to try to “compensate” the SLEκ
martingale by multiplying by a differentiable process. In the cases we consider
this does not give the correct observables, but rather corresponds to a change of
coordinates moving the target point at ∞.
3
Preliminaries
Unless specified otherwise, all complex powers are defined using the principal branch of
the logarithm, that is, z α = eα(ln |z|+iArg z) where Arg z ∈ (−π, π]. We write z = x + iy
and let
1 ∂
∂
1 ∂
∂
¯
∂=
−i
,
∂=
+i
.
2 ∂x
∂y
2 ∂x
∂y
We let H = {z ∈ C : Im z > 0} and D = {z ∈ C : |z| < 1} denote the open upper
half-plane and the open unit disk, respectively. The open disk of radius > 0 centered at
z ∈ C will be denoted by B (z) = {w ∈ C : |w − z| < }. Throughout the paper, c > 0
and C > 0 will denote generic constants which may change within a computation.
Let D be a simply connected domain with two distinct boundary points p, q (prime
ends). There is a conformal transformation f : D → H taking p to 0 and q to ∞; in fact,
f is determined only up to a final scaling. We choose one such f , but the quantities we
define do not depend on the choice. Given z ∈ D, we define the conformal radius rD (z)
of D seen from z by letting
ΥD (z) =
Im f (z)
,
|f 0 (z)|
rD (z) = 2ΥD (z).
Schwarz’ lemma and Koebe’s 1/4 theorem give the distortion estimates
dist(z, ∂D)/2 6 ΥD (z) 6 2 dist(z, ∂D).
13
(3.1)
We define
SD,p,q (z) = sin[arg f (z)],
S(z) = SH,0,∞ (z),
and note that this is a conformal invariant. Suppose D is a Jordan domain and that
J− , J+ are the boundary arcs f −1 (R− ) and f −1 (R+ ), respectively. Let ωD (z, E) denote
the harmonic measure of E in D from z. Then it is easy to see that
SD,p,q (z) min{ωD (z, J− ), ωD (z, J+ )},
(3.2)
with the implicit constants universal. By conformal invariance an analogous statement
holds for any simply connected domain different from C. We will use this relation several
times without explicitly saying so in order to estimate SD,p,q . In many places we will estimate harmonic measure using the Beurling estimate, see for example [21] Theorem IV.6.2
(with θ = 2π).
We will also make use of excursion measure. Suppose D is analytic with two disjoint
boundary arcs A, B. We define the excursion measure in D between A and B by
ED (A, B) =
Z
Z
∂n ω(ζ, A)|dζ|,
∂n ω(ζ, B)|dζ| =
B
A
where ω is harmonic measure and ∂n denotes normal derivative in the inward pointing
direction. For example, one has
π EH ([−x, 0], [1, y]) = ln y − ln
y+x
,
1+x
so that
πEH ((−∞, 0], [1, 1 + x]) = ln(1 + x) = x + O(x2 ),
(3.3)
as x ↓ 0. Excursion measure is clearly a conformal invariant, and consequently we can
define it in rough domains by mapping to the half plane and computing there.
3.1
Schramm-Loewner evolution
Let 0 < κ < 8. Throughout the paper we will use the following parameters:
a=
2
,
κ
r = rκ (ρ) =
ρ
ρa
=
,
κ
2
d=1+
1
,
4a
β = 4a − 1.
We will also sometimes write α = 4a. The assumption κ = 2/a < 8 implies that α > 1.
We will work with the κ-dependent Loewner equation
∂t gt (z) =
a
,
gt (z) − ξt
g0 (z) = z,
(3.4)
where ξt , t > 0, is the (continuous) Loewner driving term. The solution is a family
of conformal maps (gt (z)) called the Loewner chain of ξt . The SLEκ Loewner chain is
obtained by taking ξt to be a standard Brownian motion and a = 2/κ. The SLEκ path is
the continuous curve connecting 0 with ∞ in H defined by
γ(t) = lim gt−1 (ξt + iy),
y↓0
14
γt := γ[0, t].
J−
D
q
η
J+
p
Figure 3. The domain D and the crosscut η of Lemma 3.2.
We write Ht for the simply connected domain given by taking the unbounded component
of H \ γt . Given a simply connected domain D with distinct boundary points p, q, we
define chordal SLEκ in D from p to q by conformal invariance. We write
St (z) = SHt ,γ(t),∞ (z),
Υt (z) = ΥHt (z) =
Im gt (z)
.
|gt0 (z)|
We will make use of the following sharp one-point estimate which also defines the Green’s
function for chordal SLEκ , see [30] for a proof.
Lemma 3.1 (Green’s function for chordal SLEκ ). Suppose 0 < κ < 8. Let γ be SLEκ
in D from p to q, where D is a simply connected domain with distinct boundary points
(prime ends) p, q. Then there exists a constant c > 0 such that as → 0 the following
estimate holds uniformly with respect to all z ∈ D satisfying dist(z, ∂D) > 2:
P (Υ∞ (z) 6 ) = c∗ 2−d GD (z; p, q) [1 + O(c )] ,
where, by definition,
β
GD (z; p, q) = ΥD (z)d−2 SD,p,q
(z)
is the Green’s function for SLEκ from p to q in D, and c∗ is the constant defined in (2.11).
We also need to use a boundary estimate for SLE which is convenient to express in
terms of excursion measure, see Lemma 4.5 of [30].
Lemma 3.2. Let 0 < κ < 8. Suppose D is a simply connected Jordan domain and let
p, q ∈ ∂D be two distinct boundary points. Write J− , J+ for the boundary arcs connecting q
with p and p with q in the counterclockwise direction, respectively. Suppose η is a crosscut
of D starting and ending on J+ , see Figure 3. Then, if γ is chordal SLEκ in D from p to
q,
P (γ∞ ∩ η 6= ∅) 6 c ED\η (J− , η)β ,
(3.5)
where β = 4a − 1 and the constant c ∈ (0, ∞) is independent of D, p, q, and η.
15
3.1.1
SLEκ (ρ)
Let 0 < κ < 8 and ρ ∈ R. We will work with SLEκ (ρ) as defined by weighting SLEκ
by a local martingale. Let (ξ 1 , ξ 2 ) ∈ R2 be given with ξ 1 < ξ 2 . Suppose ξt1 is Brownian
motion started from ξ 1 under the measure P, with filtration Ft . We refer to P as the
SLEκ measure. Let (gt ) be the SLEκ Loewner flow defined by equation (3.4) with ξt = ξt1
and set
ξt2 := gt (ξ 2 ).
(3.6)
We call ξ 2 the force point. Define
λ(r) =
r
(r − β) ,
2a
ζ(r) = λ(−r) − r =
r
(r + 2a − 1) .
2a
Note that ζ > 0 whenever 0 < κ 6 4 and r > 0. Itô’s formula shows that
(ρ)
Mt
ξt2 − ξt1
ξ2 − ξ1
=
!r
gt0 (ξ 2 )ζ(r)
(3.7)
is a local P-martingale for any ρ ∈ R. In fact,
(ρ)
dMt
(ρ)
Mt
=
−r
dξ 1 .
ξt2 − ξt1 t
The SLEκ (ρ) measure Pρ = Pρξ1 ,ξ2 is defined by weighting P by the martingale M (ρ) , that
is,
(ρ)
Pρ (V ) = E[Mt 1V ] for V ∈ Ft .
(3.8)
Then, using Girsanov’s theorem, the equation for ξt1 changes to
dξt1 =
ξt1
r
dt + dWt ,
− ξt2
(3.9)
where Wt is Pρ -Brownian motion. This is the defining equation for the driving term of
SLEκ (ρ). (Since M (ρ) is a local martingale we need to stop the process before M (ρ) blows
up; we will not always be explicit about this. We will not need to consider SLEκ (ρ) after
the time the path hits the force point.) We refer to the Loewner chain driven by ξt1 under
Pρ as SLEκ (ρ) started from (ξ 1 , ξ 2 ). If ρ is sufficiently negative, the SLEκ (ρ) path will
almost surely hit the force point. In this case it can be useful to reparametrize so that
the quantity
Ct = Ct (ξ 2 ) =
ξt2 − Ot
,
gt0 (ξ 2 )
(3.10)
decays deterministically; this is called the radial parametrization in this context. Here Ot
is defined as the image under gt of the rightmost point in the hull at time t; in particular,
Ot = gt (0+) if 0 < κ 6 4, see, e.g., [1]. Geometrically Ct equals (1/4 times) the conformal
16
radius seen from ξ 2 in Ht after Schwarz reflection. We define a time-change s(t) so that
Ĉt := Cs(t) = e−at . A computation shows that if
At =
ξt2 − Ot
ξt2 − ξt1
then s0 (t) = (ξˆt2 − ξˆt1 )2 (Â−1
t −1), where Ât = As(t) , see e.g. Section 2.2 of [1]. An important
fact is that Ât is positive recurrent with respect to SLEκ (ρ) if ρ is chosen appropriately.
Lemma 3.3. Suppose 0 < κ < 8 and ρ < κ/2 − 4. Consider SLEκ (ρ) started from (0, 1).
Then Ât is positive recurrent with invariant density
πA (x) = c0 x−β−aρ (1 − x)2a−1 ,
c0 =
Γ(2 − 2a − aρ)
.
Γ(2a)Γ(2 − 4a − aρ)
In fact, there is α > 0 such that if f is integrable with respect to the density πA , then as
t → ∞,
Z 1
h
i
f (x) πA (x) dx 1 + O(e−αt ) .
E f (Ât ) = c0
0
Proof. See Section 2 of [1] and Section 5.2 of [28].
3.1.2
Relationship between multiple SLE and SLEκ (ρ)
Consider a system of two commuting chordal SLEs curves started from (ξ 1 , ξ 2 ); recall
(2.1) and (2.2). Suppose we first grow γ2 up to capacity t. The conditional law of gt ◦ γ1
is then an SLEκ (2) in H started from (ξt1 , ξt2 ). In particular, the marginal law of γ1 is that
of an SLEκ (2) started from ξ 1 with force point ξ 2 . Indeed, if we choose the particular
growth speeds λ1 (t) ≡ a and λ2 (t) ≡ 0, then the defining equations (2.1) and (2.2) for
commuting SLE reduce to

a

∂ g (z) = g (z)−ξ

1,

t
 t t
t
g0 (z) = z,
t
ξ 1 −ξ 2



2
dξt = 2 a 1 dt,
ξt −ξt
ξ02
dξ 1 =
a
dt + dBt1 ,
ξ01 = ξ 1 ,
=
(3.11)
ξ2,
where Bt1 is P-Brownian motion. Evaluating the equation for gt (z) at z = ξ 2 we infer
that ξt2 = gt (ξ 2 ). Comparing this with the equations (3.6) and (3.9) defining SLEκ (ρ), we
conclude that γ1 (t) has the same distribution under the commuting SLEκ measure P as
it has under the SLEκ (2)-measure P2 started from (ξ 1 , ξ 2 ).
3.1.3
Two-sided radial SLE and radial parametrization
Recall that if z ∈ H is fixed then the SLEκ Green’s function in Ht equals
Gt = Gt (z) = Υd−2
(z)Stβ (z),
t
(3.12)
which is a covariant P-martingale. Two-sided radial SLE in H through z is the process
obtained by weighting chordal SLEκ by Gt . (This is the same as SLEκ (κ − 8) with force
17
point z ∈ H.) Since two-sided radial SLE approaches its target point, it is natural to use
the radial parametrization, so that the conformal radius (seen from z) decays deterministically. More precisely, we change time so that Υs(t) (z) = e−2at ; this parametrization
depends on z. The Loewner equation implies
d ln Υt = −2a
yt2
dt,
|zt |4
zt = xt + iyt = gt (z) − ξt1 .
Hence s0 (t) = |z̃t |4 /ỹt2 , where S̃t = Ss(t) , z̃t = zs(t) , etc., denote the time-changed processes.
Using that
yt
xt yt
dξ 1 ,
dΘt = (1 − 2a) 4 dt +
|zt |
|zt |2 t
we find that Θ̃t = Θs(t) satisfies
dΘ̃t = (1 − 2a) cot Θ̃t dt + dW̃t ,
where W̃t is standard P-Brownian motion. The time-changed martingale can be written
G̃t = e−2a(d−2)t S̃tβ .
(3.13)
The two-sided radial SLEκ measure P∗ = P∗z is defined by weighting chordal SLEκ by G̃t ,
that is,
P∗ (V ) = G̃−1
0 E[G̃t 1V ],
V ∈ F̃t .
(3.14)
This produces two-sided radial SLEκ in the radial parametrization.
Since dG̃t = β G̃t cot(Θ̃t )dW̃t , Girsanov’s theorem implies that the equation for Θ̃t
changes to the radial Bessel equation under the new measure P∗ :
dΘ̃t = 2a cot Θ̃t dt + dB̃t ,
where B̃t is P∗ -standard Brownian motion.
We will use the following lemma about the radial Bessel equation, see, e.g., Section 3
of [29].
Lemma 3.4. Let 0 < κ < 8, a = 2/κ and suppose the process (Θt ) is a solution to the
SDE
dΘt = 2a cot Θt dt + dBt , Θ0 = Θ.
(3.15)
Then Θt is positive recurrent with invariant density
ψ(x) =
c∗
sin4a x,
2
where c∗ is the constant in (2.11). In fact, there is α > 0 such that if f is integrable with
respect to the density ψ, then as t → ∞,
Z π
E [f (Θt )] =
f (x) ψ(x) dx (1 + O(e−αt )),
0
where the error term does not depend on Θ0 .
18
4
4.1
Martingale observables as CFT correlation functions
Screening
The CFT framework of Kang and Makarov [23] can be used to generate martingale observables for SLE systems, see in particular Lecture 14 of [23]. The framework of [23] has been
extended incorporate several commuting SLEs started from different points in [3]. We will
use the screening method [13] which produces observables in the form of contour integrals,
which we call screening integrals or Coulomb gas integrals. Let us briefly and informally
describe the method. From the CFT perspective (we work in the framework of [23]), one
starts from a CFT correlation function with appropriate field insertions giving a corresponding (known) SLEκ martingale. Adding additional paths means inserting additional
boundary fields. This will create observables for the system of SLEs. But in the cases we
consider the extra fields change the boundary behavior so that the new observable does
not encode the desired geometric information anymore. To remedy this, carefully chosen
auxiliary fields are insterted and then integrated out along integration contours. The
correct choices of insertions and integration contours depend on the particular problem,
and different choices correspond to solutions with different boundary behavior.
Remark 4.1. We mention in passing that from a different point of view, it is known that
the Gaussian free field with suitable boundary data can be coupled with SLE paths as
“local sets” for the field [31]. By the nature of the coupling, correlation functions for the
field should give rise to SLE martingales. The appropriate boundary conditions can be
understood as insertions of suitable fields on the boundary.
In what follows, we briefly summarize how we used these ideas to arrive at the explicit
expressions (2.5) and (2.8) for the Schramm probability P (z, ξ) and the Green’s function
G(z, ξ 1 , ξ 2 ), respectively. Since the discussion is purely motivational, except for Lemma 4.2
we make no attempt in this section to be complete or rigorous (this is in contrast to the
other sections of the paper which are rigorous). We refer to [23, 3] for an introduction to
the underlying CFT framework and we will use notation from this reference.
Consider a system of two commuting SLEs started from (ξ 1 , ξ 2 ) ∈ R2 . If λ1 (t) and
λ2 (t) denote the growth speeds of the two curves, the evolution of the system is described
by equations (2.1) and (2.2). In the conformal field theory language of [23], the presence
of two commuting SLE curves in H started from ξ 1 and ξ 2 corresponds to the insertion of
the operator
√
√
i a
i a
O(ξ 1 , ξ 2 ) = V?,(b) (ξ 1 )V?,(b) (ξ 2 ),
iσ (z) denotes a rooted vertex field inserted at z (see[23], p. 96) and the parameter
where V?,(b)
b satisfies the relation
√ √
2 a( a + b) = 1,
a = 2/κ.
(4.1)
p
Notice that we define a = 2/κ while [23] defines “a” by 2/κ. The framework of [23]
(or rather an extension of this framework to the case of multiple curves [3]) suggests
that if {zj }n1 ⊂ C are points and {Xj }n1 are fields satisfying certain properties, then the
correlation function
(z1 ,...,zn )
Mt
−1
−1
= ÊH
O(ξ 1 ,ξ 2 ) [(X1 ||gt )(z1 ) · · · (Xn ||gt )(zn )]
t
t
19
(4.2)
is a (local) martingale observable for the system when evaluated in the “Loewner charts”
(gt ). It turns out that the observables relevant for Schramm’s formula and for the Green’s
function belong to a class of correlation functions of the form
(z,u)
Mt
iσ1
iσ2
is
= ÊO(ξt ) [(V?,(b)
||gt−1 )(z)(V?,(b)
||gt−1 )(z)(V?,(b)
||gt−1 )(u)],
(4.3)
where z ∈ H, u ∈ C, and σ1 , σ2 , s ∈ R are real constants. We will later integrate out the
is ||g −1 )(u) in the definition
variable u, but it is essential to include the screening field (V?,(b)
t
(4.3) in order to arrive at observables with the appropriate conformal dimensions at z and
(z,u)
at infinity. The observable Mt
can be written as
(z,u)
Mt
gt0 (z)
=
2
σ1
−σ1 b
2
gt0 (z)
2
σ2
−σ2 b
2
s2
gt0 (u) 2 −sb A(Zt , ξt1 , ξt2 , Ut ),
(4.4)
where Zt = gt (z), Ut = gt (u), and the function A(z, ξ 1 , ξ 2 , u) is defined by
A(z, ξ 1 , ξ 2 , u) = (z − z̄)σ1 σ2 (z − ξ 1 )(z − ξ 2 )
σ1 √a (z̄ − ξ 1 )(z̄ − ξ 2 )
× (z − u)σ1 s (z̄ − u)σ2 s (u − ξ 1 )(u − ξ 2 )
s√a
.
(z,u)
The following lemma confirms that the CFT generated observable Mt
martingale for any choice of z, u ∈ H and σ1 , σ2 , s ∈ R.
σ2 √a
(4.5)
is indeed a local
Lemma 4.2. Let z, u ∈ H be two distinct points in the upper half-plane and let σ1 , σ2 , s ∈
R be real numbers. For any choice of the growth speeds λj (t), j = 1, 2, the function
(z,u)
Mt
defined in (4.4) is a local martingale for the system of commuting SLEs started
from (ξ 1 , ξ 2 ).
Proof. It is enough to show that the drift coefficient Dt in the expression
(z,u)
dMt
(z,u)
vanishes. Writing M := Mt
imply
Dt =
= Dt dt + Et1 dBt1 + Et2 dBt2
, equations (2.1) and (2.2) together with Itô’s formula
∂M dZt ∂M dZ̄t ∂M dUt
∂M ∂gt0 (z)
∂M ∂gt0 (z)
∂M ∂gt0 (u)
+
+
+ 0
+ 0
+ 0
∂Zt dt
∂Ut dt
∂gt (z) ∂t
∂gt (u) ∂t
∂ Z̄t dt
∂gt (z) ∂t
+
2
X
∂M ∂ξtj
j=1
∂ξtj ∂t
+
2
κX
∂2M
λj (t)
,
4 j=1
∂ 2 ξtj
(4.6)
where Zt = Xt + iYt and
2
X
dZt
λj (t)
=
j,
dt
j=1 Zt − ξt
2
X
dZ̄t
λj (t)
=
j,
dt
j=1 Z̄t − ξt
2
X
∂gt0 (z)
gt0 (z)λj (t)
=−
j 2,
∂t
j=1 (Zt − ξt )
2
X
dUt
λj (t)
=
j,
dt
j=1 Ut − ξt
2
X
∂gt0 (z)
gt0 (z)λj (t)
=−
j 2,
∂t
j=1 (Z̄t − ξt )
20
2
X
∂gt0 (u)
gt0 (u)λj (t)
=−
j 2,
∂t
j=1 (Ut − ξt )
∂ξt1
λ1 (t) + λ2 (t)
,
=
∂t
ξt1 − ξt2
∂ξt2
λ1 (t) + λ2 (t)
.
=
∂t
ξt2 − ξt1
(4.7)
Substituting the expressions in (4.7) into (4.6), a direct computation shows that Dt =
0.
Since (4.4) is a local martingale for each value of the screening variable u, we expect
the integrated observable
(z)
Mt
Z
=
γ
(z,u)
Mt
du
(4.8)
to be a local martingale for any choice of z ∈ H, σ1 , σ2 , s ∈ R, and of the integration
contour γ, at least as long as the integral in (4.8) converges and the branches of the
complex powers in (4.5) are consistently defined. The integral in (4.8) is referred to as a
“screening” integral.
(z)
By choosing λ2 (t) = 0 in Lemma 4.2, we see that the observable Mt defined in
(4.8) is a local martingale for SLEκ (2) started from (ξ 1 , ξ 2 ). We next describe how the
martingales relevant for Schramm’s formula and for the Green’s function for SLEκ (2) arise
(z)
as special cases of Mt corresponding to particular choices of σ1 , σ2 , s ∈ R and of the
contour γ.
4.2
Prediction of Schramm’s formula
In order to obtain the local martingale relevant for Schramm’s formula we choose the
following values for the parameters (“charges”) in (4.4):
√
√
σ1 = −2 a,
σ2 = 2b,
s = −2 a.
(4.9)
√
The choice (4.9) can be motivated as follows. First of all, by choosing s = −2 a we
(z)
ensure that s2 /2 − sb = 1 (see (4.1)). This implies that Mt involves the one-form
2
gt0 (u)s /2−sb du = gt0 (u)du. After integration with respect to du this leads to a conformally
invariant screening integral. To motivate the choices of σ1 and σ2 , let P (z, ξ 1 , ξ 2 ) denote
the probability that the point z ∈ H lies to the left of an SLEκ (2)-path started from
(ξ 1 , ξ 2 ). Then we expect ∂z P to be a martingale observable with conformal dimensions
λ(z) = 1,
λ∗ (z) = 0,
λ∞ = 0.
(4.10)
(z)
The parameters in (4.9) are chosen so that the observable Mt in (4.8) has the conformal
dimensions in (4.10). We emphasize that it is the inclusion of the screening field in (4.3)
that makes it possible to obtain these dimensions. In particular, by including it we can
have λ∞ = 0. We have considered the derivative ∂z P instead of P because then we are
able to construct a nontrivial martingale with the correct dimensions.
In the special case when the parameters σ1 , σ2 , s are given by (4.9), the local martingale
(4.8) takes the form
(z)
Mt
α
α
α
α
= gt0 (z)(Zt − Z̄t )α−2 (Zt − ξt1 )− 2 (Zt − ξt2 )− 2 (Z̄t − ξt1 )1− 2 (Z̄t − ξt2 )1− 2
21
×
Z
γ
(Zt − Ut )α (Z̄t − Ut )α−2 (Ut − ξt1 )(Ut − ξt2 )
− α 0
2
gt (u)du.
(4.11)
We expect from the above discussion that there exists an appropriate choice of the inte(z)
gration contour γ in (4.8) such that ∂z P (z, ξ 1 , ξ 2 ) = const × M0 , that is, we expect
α
α
α
α
∂z P (z, ξ 1 , ξ 2 ) = c(κ)y α−2 (z − ξ 1 )− 2 (z − ξ 2 )− 2 (z̄ − ξ 1 )1− 2 (z̄ − ξ 2 )1− 2
×
Z
α
α
(u − z)α (u − z̄)α−2 (u − ξ 1 )− 2 (u − ξ 2 )− 2 du,
γ
where c(κ) is a complex constant. Setting ξ 1 = 0 and ξ 2 = ξ in this formula, we arrive
at the prediction (2.5) for the Schramm probability P (z, ξ). Indeed, the integration with
respect to x in (2.5) recovers P from ∂z P and ensures that P tends to zero as Re z →
∞. On the other hand, the choice of the integration contour from z̄ to z in (2.4) is
mandated by the requirement that P (z, ξ) should satisfy the correct boundary conditions
as z approaches the real axis. Finally, P (z, ξ) must be a real-valued function tending to
1 as Re z → −∞; this fixes the constant c(κ).
4.3
Prediction of the Green’s function
In order to obtain the local martingale relevant for the SLEκ (2) Green’s function, we
choose the following values for the parameters in (4.4):
√
√
√
σ1 = b − a,
σ2 = b − a,
s = −2 a,
(4.12)
√
(z)
As in the case of Schramm’s formula, the choice s = −2 a ensures that Mt involves the
one-form gt0 (u)du. Moreover, if we let G(z, ξ 1 , ξ 2 ) denote the Green’s function for SLEκ (2)
started from (ξ 1 , ξ 2 ), then we expect G to have the conformal dimensions (cf. page 124 in
[23])
λ(z) = λ∗ (z) =
2−d
,
2
λ∞ = 0.
(4.13)
(z)
The parameters σ1 and σ2 in (4.12) are determined so that the observable Mt in (4.8)
has the conformal dimensions in (4.13). For example, a generalization of Proposition 15.5
√
2
in [23] to the case of two curves implies that λ∞ = (2 a − b)Σ + Σ2 = 0 where Σ =
√
σ1 + σ2 − 2 a.
Remark 4.3. We can see here that the choice ρ = 2 is special: we have only two possible
√
√
ways to add one screening field, corresponding to s =
√−2 a or s = 1/ a. But√the extra
ρ = 2 corresponds to additional charges σ = σ∗ = 2/ 8κ (we are using σ = ρ/ 8κ, so at
√
infinity we have an additional charge σ + σ∗ = 2 a. We see that the ρ = 2 charge can be
screened by only one screening field. We can also see that if we add more ρ:s, they can be
√
screened if their charges sum up to 2 a. Also this suggests that every SLEκ observable
with λq = 0 gives an SLEκ (2) observable with λq = 0 after screening. Simlarly, since
√
adding n additional ρj = 2 gives additional charges at ∞ of 2n a, one could expect that
one can construct a martingale for a system of n SLEs by adding n screening charges.
22
In the special case when the parameters σ1 , σ2 , s are given by (4.12), the local martingale (4.8) takes the form
(z)
Mt
= |gt0 (z)|2−d
Z
γ
A(Zt , ξt1 , ξt2 , gt (u))gt0 (u)du,
(4.14)
where
1
A(z, ξ 1 , ξ 2 , u) = (z − z̄)α+ α −2 |z − ξ 1 |−β |z − ξ 2 |−β
× (z − u)β (z̄ − u)β (u − ξ 1 )(u − ξ 2 )
− α
2
.
(4.15)
We expect from the above discussion that there exists an appropriate choice of the inte(z)
gration contour γ in (4.8) such that G(z, ξ 1 , ξ 2 ) = const × M0 , that is, we expect
1
G(z, ξ 1 , ξ 2 ) = c(κ)y α+ α −2 |z − ξ 1 |−β |z − ξ 2 |−β J(z, ξ 1 , ξ 2 ),
where
1
2
Z
J(z, ξ , ξ ) =
α
(4.16)
α
(u − z)β (u − z̄)β (u − ξ 1 )− 2 (ξ 2 − u)− 2 du
γ
and c(κ) is a complex constant. By requiring that G satisfy the correct boundary conditions, we arrive at the prediction (2.8) for the Green’s function for SLEκ (2). The trickiest
step is the determination of the appropriate screening contour γ. This contour must
be chosen so that the Green’s function satisfies the appropriate boundary conditions as
(z, ξ 1 , ξ 2 ) approach the boundary of the domain H × {−∞ < ξ 1 < ξ 2 < ∞}. The complete verification that the Pochhammer integration contour in (2.4) leads to the correct
boundary behavior is presented in Lemma 6.2 and relies on a complicated analysis of integral asymptotics. We first arrived at the Pochhammer contour in (2.4) via the following
simpler argument.
Let Gξ (z) = G(z, −ξ, ξ), Jξ (z) = J(z, −ξ, ξ). Let also Iξ (z) = I(z, −ξ, ξ) where I is
the function defined in (2.7), i.e.,
Z (z+,ξ+,z−,ξ−)
Iξ (z) =
α
α
(u − z)α−1 (u − z̄)α−1 (ξ + u)− 2 (ξ − u)− 2 du.
(4.17)
A
We make the ansatz that
Jξ (z) =
4
X
i=1
Z
ci (κ)
α
α
(u − z)α−1 (u − z̄)α−1 (ξ + u)− 2 (ξ − u)− 2 du,
γi
where the contours {γi }41 are Pochhammer contours surrounding the pairs (ξ, z), (ξ, z̄),
(−ξ, z), and (−ξ, z̄), respectively. The integral involving the pair (ξ, z) is Iξ (z). The
integrals involving the pairs (±ξ, z) are related via complex conjugation to the integrals
involving the pairs (±ξ, z̄). Moreover, by performing the change of variables u → −ū,
we see that the integral involving the pair (−ξ, z) can be expressed in terms of I(−z̄).
Thus, using the requirement that J(z, ξ) be real-valued, we can without loss of generality
assume that J(z, ξ) is a real linear combination of the real and imaginary parts of Iξ (z) and
Iξ (−z̄).
23
At this stage it is convenient, for simplicity, to assume 4 < κ < 8 so that 1 < α < 2.
Then we can collapse the contour in the definition (4.17) of Iξ (z) onto a curve from ξ from
z; this gives
Iξ (z) = (1 − e2iπα + eiπα − e−iπα )Iˆξ (z),
where Iˆξ (z) is defined by
Iˆξ (z) =
Z z
α
α
(u − z)α−1 (u − z̄)α−1 (ξ + u)− 2 (ξ − u)− 2 du.
ξ
Since Iˆ obeys the symmetry Im Iˆξ (z) = Im Iˆξ (−z̄), our ansats takes the form
Jξ (z) = A1 Re Iˆξ (z) + A2 Re Iˆξ (−z̄) + A3 Im Iˆξ (z),
(4.18)
where Aj = Aj (κ), j = 1, 2, 3, are real constants.
Up to factors which are independent of y, we expect the Green’s function Gξ (z) to
satisfy
1
Gξ (x + iy) ∼ y d−2 = y α −1 ,
Gξ (ξ + iy) ∼ y
d−2 β+2a
y
=y
y → ∞,
1
+ 3α
−2
α
2
,
x fixed,
(4.19a)
y ↓ 0.
(4.19b)
Indeed, since the influence of the force point ξ 2 goes to zero as Im γ(t) becomes large,
the first relation follows by comparison with SLEκ . The second relation can be motivated
by noticing that the boundary exponent for SLEκ (ρ) at the force point ξ 2 is β + ρa, see
Lemma 7.3. In terms of Jξ (z), the estimates (4.19) translate into
Jξ (x + iy) ∼ y α−1 ,
Jξ (ξ + iy) ∼ y
3α
−1
2
y → ∞,
x fixed,
(4.20a)
y ↓ 0.
,
(4.20b)
We will use these conditions to fix the values of the Aj ’s.
We obtain one constraint on the Aj ’s by considering the asymptotics of Jξ (iy) as
y → ∞. Indeed, for x = 0 we have
Iˆξ (iy) =
Z iy
α
(u2 + y 2 )α−1 (ξ 2 − u2 )− 2 du
ξ
i √ −α 2α−2 yΓ(α)
πξ y
=
2 F1
2
Γ(α + 12 )
iξΓ(1 − α2 )
+
2 F1
Γ( 32 − α2 )
1 α
1 y2
, ,α + ,− 2
2 2
2 ξ
3 α ξ2
1
, 1 − α, − , − 2
2
2
2
y
!
!
,
where 2 F1 denotes the standard hypergeometric function. This implies
Iˆξ (iy) = y
α−1
+y
iΓ( 21 − α2 )Γ(α)
1
+O
2
y
2Γ( α+1
)
2
2(α−1)
!
πα
π 3/2 ξ 1−α (csc( πα
1
2 ) + i sec( 2 ))
−
+O
3
α
α
y2
2(Γ( 2 − 2 )Γ( 2 ))
24
!
,
y → ∞.
Substituting this expansion into (4.18), we find an expression for Jξ (iy) involving two
terms which are proportional to y 2(α−1) and y α−1 , respectively, as y → ∞. In order to
satisfy the condition (4.20a), we must choose the Aj so that the coefficient of the larger
term involving y 2(α−1) vanishes. This leads to the relation
A1 + A2
πα
= − tan
.
A3
2
(4.21)
We obtain a second constraint on the Aj ’s by considering the asymptotics of Jξ (iy) as
z → ξ. Indeed, for x = ξ we have
Z y
iπ
α
Iˆξ (ξ + iy) = e 2 (1+ 2 )
α
(y 2 − s2 )α−1 (2ξ + is)α−1 s− 2 ds.
0
Hence
iπ
α
α
Iˆξ (ξ + iy) ∼ e 2 (1+ 2 ) (2ξ)− 2
−α
−1
2
2
e
1
iπ(α+2)
4
=
Γ
Z y
α
(y 2 − s2 )α−1 s− 2 ds
0
−α/2
ξ
Γ 21
3α
4
1
2
+
−
α
4
Γ(α)
y
3α
−1
2
y ↓ 0,
,
ξ > 0.
(4.22)
Similarly, for x = −ξ, we have
ˆ
I(−ξ
+ iy, ξ) =
Z −ξ
α
α
((u + ξ)2 + y 2 )α−1 (ξ + u)− 2 (ξ − u)− 2 du
ξ
Z y
+
α
α
(i(s − y))α−1 (i(s + y))α−1 (is)− 2 (2ξ − is)− 2 ids.
0
Hence
ˆ
I(−ξ
+ iy, ξ) ∼ −
Z ξ
(u + ξ)
3α
−2
2
−α
2
(ξ − u)
du + e
iπ
(1− α
)
2
2
(2ξ)
−ξ
−α
2
Z y
α
(y 2 − s2 )α−1 s− 2 ds
0

= 2ξ α−1 
α
2 F1 1, 2 −
3α
2 ,2
−
α−2
1
i2− 2 −1 e− 4 iπα ξ −α/2 Γ
+
Γ
3α
4
+
1
2
1
2
−
α
2 ; −1
α
4
+
2 F1 1,
α 3α
2 , 2 ; −1
2 − 3α
Γ(α)
y
3α
−1
2
,
y ↓ 0,


ξ > 0.
(4.23)
Substituting the expansions (4.22) and (4.23) into (4.18), we find an expression for Jξ (ξ +
3α
iy) involving two terms which are of order O(y 2 −1 ) and O(1), respectively, as y → 0. In
order to satisfy the condition (4.20b), we must choose the Aj so that the coefficient of the
larger term of O(1) vanishes. This implies
A2 = 0.
(4.24)
Using the constraints (4.21) and (4.24), the expression (4.18) becomes
Jξ (z) = B1 Im e−
iπα
2
Iˆξ (z) = B2 Im e−iπα Iξ (z) ,
25
where Bj = Bj (κ), j = 1, 2, are real constants. Recalling (4.16), this gives the following
expression for Gξ (z) = G(z, −ξ, ξ):
1
1
G(z, −ξ, ξ) = y α+ α −2 |z + ξ|1−α |z − ξ|1−α Im e−iπα I(z, −ξ, ξ) ,
ĉ
z ∈ H,
ξ > 0,
where ĉ(κ) is an overall real constant yet to be determined. Using translation invariance to
extend this expression to an arbitrary starting point (ξ 1 , ξ 2 ), we find (2.8). The derivation
here used that 4 < κ < 8, but by analytic continuation we expect the same formula to
hold for 0 < κ 6 4.
Remark 4.4. We remark here that the non-screened martingale obtained via Girsanov
has the conformal dimensions
λ(z) = λ∗ (z) =
5
2−d
,
2
λ∞ = −β.
(4.25)
Schramm’s formula
This section proves Theorem 2.1. The strategy is the same as in Schramm’s original
argument [35]. Assume 0 < κ < 8, i.e., α = 8/κ > 1. We write the function M(z, ξ)
defined in (2.4) as
α
α
α
α
M(z, ξ) = y α−2 z − 2 (z − ξ)− 2 z̄ 1− 2 (z̄ − ξ)1− 2 J(z, ξ),
z ∈ H,
ξ > 0,
(5.1)
z ∈ H,
ξ > 0,
(5.2)
where J(z, ξ) is defined by
Z z
J(z, ξ) =
α
α
(u − z)α (u − z̄)α−2 u− 2 (u − ξ)− 2 du,
z̄
and the contour from z̄ to z passes to the right of ξ as in Figure 1. We want to prove that
the probability that the system started from (0, ξ) passes to the right of z is given by
P (z, ξ) =
1
cα
Z ∞
Re M(x0 + iy, ξ)dx0 ,
x
x ∈ R,
y > 0,
ξ > 0.
The idea is to apply Itô’s formula and a stopping time argument to prove that the prediction is correct. In the present situation the hard work lies in verifying that the function
P (z, ξ) has sufficient regularity for Itô’s formula to apply and that it satisfies the appropriate boundary conditions. Most of the needed analysis is carried out in Appendix A
and the argument presented in this section uses results from there. Once we have proved
Theorem 2.1, we easily obtain fusion formulas by simply collapsing the seeds.
5.1
Proof of Theorem 2.1
Let P (z, ξ) be the function defined in (2.5). In Appendix A, we carefully analyze the
function P (z, ξ) and show that it is well-defined, smooth, and fulfills the correct boundary
conditions. We summarize these facts here and then use them to give the short proof of
Theorem 2.1.
26
Lemma 5.1. The function P (z, ξ) defined in (2.5) is a well-defined smooth function of
(z, ξ) ∈ H × (0, ∞).
Proof. See Lemma A.8.
Lemma 5.2. We have
|P (z, ξ)| 6 C (arg z)α−1 ,
z ∈ H,
|P (z, ξ) − 1| 6 C (π − arg z)
α−1
,
ξ > 0,
z ∈ H,
(5.3a)
ξ > 0.
(5.3b)
Proof. See Lemma A.12.
Consider a system of commuting SLEs in H started from 0 and ξ > 0, respectively.
We grow both paths simultaneously with some given speeds λj (t) > 0. Write ξt1 and ξt2
for the Loewner driving terms of the system and let gt denote the solution of (2.1) which
uniformizes the whole system at capacity t. Then ξt1 and ξt2 are the images of the tips of
the two curves under the conformal map gt . Given a point z ∈ H, let Zt = gt (z) and let
τ (z) denote the time that Im gt (z) first reaches 0.
Remark 5.3. A point z ∈ H lies to the left of both curves iff it
the leftmost curve γ1 starting from 0. Moreover, since the system
distribution is independent of the order at which the two curves are
may as well assume that λ1 (t) = 1 and λ2 (t) = 0, but this assumption
lies to the left of
is commuting, its
grown. Hence we
is not essential.
Lemma 5.4. Let z ∈ H. Define Pt (z) by
Pt (z) = P (Zt − ξt1 , ξt2 − ξt1 ),
0 6 t < τ (z).
Then Pt (z) is a martingale for the system of commuting SLEs.
Proof. See Lemma A.13.
Lemma 5.5. Let z ∈ H, and Θ1t = arg(Zt − ξt1 ). Then,
lim Θ1t = 0
t↑τ (z)
resp. lim Θ1t = π ,
t↑τ (z)
if and only if z lies to the right (resp. left) of the curve γ1 starting at 0.
Proof. See the proof of Lemma 3 in [35].
Lemma 5.6. Let P̃ (z, ξ) be the probability that the point z ∈ H lies to the left of the two
curves starting at 0 and ξ > 0, respectively. Then P̃ (z, ξ) = P (z, ξ), where P (z, ξ) is the
function defined in (2.5).
Proof. By Lemma 5.5, the angle Θ1t = arg(Zt − ξt1 ) approaches π as t ↑ τ (z) on the event
that z ∈ H lies to the left of both curves. But (5.3b) shows that
|Pt (z) − 1| = |P (Zt − ξt1 , ξt2 − ξt1 ) − 1| 6 C (π − Θ1t )α−1 ,
27
z ∈ H,
t ∈ [0, τ (z)).
Consequently, on the event that z lies to the left of both curves, Pt (z) → 1 as t ↑ τ (z). A
similar argument relying on (5.3a) shows that on the event that z ∈ H lies between or to
the right of the two curves, then Pt (z) → 0 as t ↑ τ (z).
Let τn (z) be the stopping time defined by
sin Θ1t
τn (z) = inf t > 0 :
1
6
.
n
Since Pt (z) is a martingale, we have
h
i
P0 (z) = E Pτn (z) (z) ,
z ∈ H.
n = 1, 2, . . . ,
By using the dominated convergence theorem,
h
i
lim E Pτn (z) (z) = P̃ (z, ξ).
n→∞
Since P0 (z) = P (z, ξ), this concludes the proof of the lemma and of Theorem 2.1.
5.2
The special case κ = 4
If α = κ8 > 1 is an integer, the integral (5.2) defining J(z, ξ) can be computed explicitly.
However, the formulas quickly get very complicated as α increases. In this subsection, we
consider the simplest case of α = 2 (i.e. κ = 4). We remark that this case is particularly
simple for one curve as well; indeed, the probability that an SLE4 path passes to the right
of z equals (arg z)/π.
Proposition 5.7. Let κ = 4. Then the function P (z, ξ) in (2.5) is given explicitly by
x−ξ
πξ − 2ξ arctan
+ 2y
y
x−ξ
+ π 2 ξ + (4y − 2πξ) arctan
,
z = x + iy ∈ H,
y
1
P (z, ξ) = 2
4π ξ
x
− 2 arctan
y
ξ > 0.
(5.4)
Proof. Let α = 2. Then cα = −2π 2 and an explicit evaluation of the integral in (5.2) gives
J(z, ξ) = 2iy +
2i
(z − ξ)2 arg(z − ξ) − z 2 arg z ,
ξ
z ∈ H,
ξ > 0.
Using that
arg z =
π
x
− arctan
2
y
arg(z − ξ) =
and
π
x−ξ
− arctan
,
2
y
it follows that the function M in (2.4) can be expressed as
2i
π
x−ξ
(z − ξ)2
− arctan
z(z − ξ)ξ
2
y
M(z, ξ) =
28
− z2
π
x
− arctan
2
y
+ ξy
for z = x + iy ∈ H and ξ > 0. Taking the real part of this expression and integrating with
respect to x, we find that the function P (z, ξ) in (2.5) is given by
1 ∞
Re M(x0 + iy, ξ)dx0
P (z, ξ) =
cα x
1
π
π
x0 − ξ
x0
= 2 (2y − πξ)
− (πξ + 2y)
− arctan
− arctan
2π ξ
2
y
2
y
x0 x0 − ξ ∞
.
− 2ξ arctan
arctan
y
y
x0 =x
Z
The expression (5.4) follows.
Remark 5.8. In the fusion limit, equation (5.4) is consistent with the results of [20].
Indeed, in the limit ξ ↓ 0 the expression (5.4) for P (z, ξ) reduces to
P (z, 0+) =
1
arctan t (arctan t)2
1
− 2
−
+
,
4 π (1 + t2 )
π
π2
t :=
x
,
y
which is equation (25) in [20].
6
Green’s function
In this section we prove Theorem 2.6. We recall from the discussion in Section 2 that the
proof breaks down into proving Propositions 2.12 and 2.13. Proposition 2.12 establishes
existence of a Green’s function for SLEκ (ρ) and provides a representation for this Green’s
function in terms of an expectation with respect to two-sided radial SLE. Proposition 2.13
then shows that the CFT prediction Gξ (z) defined in (2.8) obeys this representation in
the case of ρ = 2.
6.1
Existence of the Green’s function: Proof of Proposition 2.12
Let 0 < κ 6 4 and 0 6 ρ < 8 − κ and consider SLEκ (ρ) started from (ξ 1 , ξ 2 ) with ξ 1 < ξ 2 .
We recall our parameters
a = 2/κ,
r = ρa/2 = ρ/κ,
ζ(r) =
r
(r + 2a − 1) ,
2a
and the normalized local martingale
(ρ)
Mt
=
ξt2 − ξt1
ξ2 − ξ1
!r
gt0 (ξ 2 )ζ(r)
by which we can weight SLEκ in order to get SLEκ (ρ), see Section 3. We will need a
geometric regularity estimate. In order to state it, let z ∈ H and 0 < 1 < 2 < Im z. Let
γ : (0, 1] → H be a simple curve such that
γ(0+) = 0,
|γ(1) − z| = 1 ,
29
|γ(t) − z| > 1 , t ∈ [0, 1).
Write H = H \ γ where γ = γ[0, 1]. For > 0 let B be the disk of radius about z and
let U be the connected component containing z of B2 ∩ H. The set ∂B2 ∩ ∂U consists
of crosscuts of H. There is a unique one which separates z from ∞ in H and we denote
this crosscut
` = `(z, γ, 2 ).
(6.1)
See Figure 4.
Lemma 6.1. Let 0 6 κ 6 4. There exists C < ∞ and α > 0 such that the following
holds. Let z ∈ H and 0 < 1 < 2 < Im z. For > 0 define the stopping time
τ = τ = inf{t > 0 : |γ(t) − z| 6 }.
If
λ = λ1 ,2 = inf{t > τ1 : γ(t) ∩ ` 6= ∅},
where
` = `(z, γτ1 , 2 )
is as in (6.1), then on the event {τ1 < ∞}, for 0 < < 1 ,
P λ < τ < ∞ | γτ1 6 C
1
2−d 1
2
β/2
,
where β = 4a − 1.
Proof. Write B1 = B(z, 1 ), B2 = B(z, 2 ) and τ1 = τ1 , τ2 = τ2 . Given γτ1 we consider
the separating crosscut ` = `(z, γτ1 , 2 ). Let σ = max{t 6 τ1 : γ(t) ∈ `}, which is not a
stopping time but almost surely ` is a crosscut of Hσ which separates z from ∞. Write V
for the simply connected component containing z of Hσ \ `. Because one of the endpoints
of ` is the tip γ(σ), gσ (∂V \ `) − Wσ is a bounded open interval I contained in either
the positive or negative real axis. Almost surely, the curve γ 0 = γ[σ, λ] is a crosscut of
V starting and ending in `. Note that gσ (γ 0 ) − Wσ is a curve in H connecting 0 with
gσ (`) − Wσ , the latter which is a crosscut of H separating I and gσ (z) − Wσ from ∞ in
H. Therefore, if d = dist(γλ , z) 6 1 , we can use the Beurling estimate to see that
d
2
Sλ (z) 6 C
1/2
.
Consequently on the event that τ1 < ∞ and d > 2, Lemma 3.1 shows that
2−d P (τ < ∞ | γλ ) 6 C
d
d
2
β/2
6C
1
2−d 1
2
β/2
.
The last estimate uses that β/2 − (2 − d) > 0 when κ 6 4 and that d 6 1 .
On the event that τ1 < ∞ and < d 6 2 we can use Lemma 3.2 (and the Beurling
estimate to estimate the excursion measure) to see that
P (τ < ∞ | γλ ) 6 C
2
β/2
6C
the last estimate uses again that β/2 − (2 − d) > 0.
30
1
2−d 1
2
β/2
;
γ
V
2
1
z
γ(τ1 )
γ0
`
γ(σ)
γ(λ)
0
Figure 4. Schematic picture of the curve γ (solid), the open set V (shaded), and the
crosscut ` defined in (6.1). If the path reenters V and hits B(z, ) the “bad” event that
λ < τ < ∞ occurs. The probability of this event is estimated in Lemma 6.1.
Proof of Proposition 2.12. We may without loss of generality assume ξ 1 = 0 and |z| = 1.
Constants are allowed to depend on z and ξ 2 . For > 0, let
τ = τ = inf{s > 0 : Υs 6 },
τ0 = inf{s > 0 : |γ(s) − z| = },
and
λ = inf{s > τ01/2 : γ(t) ∩ ` 6= ∅},
where ` = `(z, γτ 0
1/2
, 1/4 ) is the separating crosscut as in Lemma 6.1; we are assuming is sufficiently small so that 1/4 < Im z. Let E = E be the “good” event that τ < λ.
We first claim that
lim d−2 Pρ (τ < ∞, E c ) = 0.
(6.2)
↓0
To see this, for k = 1, . . .,
σk = inf {|γ(t)| > 2k },
t>0
and
Uk = {σk−1 6 τ < σk }.
Using (3.7), we then have
h
i
Pρ (τ < ∞, E c ) = E Mτ(ρ) 1τ <∞ 1E c 6 P(τ < ∞, E c ) + C
∞
X
2kr P (τ < ∞, E c , Uk ) .
k=1
(6.3)
31
The first term on the right is o(2−d ) using Lemma 6.1. We will estimate the series. For
this, suppose j = −1, . . . , J = dlog2 (−1 )e and define
0
0
Vjk = {τ2
j−1 6 σk−1 < τ2j }.
We can write
J
X
c
P (τ < ∞, E , Uk ) =
P τ < ∞, E c , Uk , Vjk .
(6.4)
j=−1
Let us first assume j 6
log2 −1 . We claim that on the event Vjk ∩ {σk−1 < ∞},
1
2
P τ < ∞, E c | γσk−1 6 P τ < ∞ | γσk−1 6 C
j β/2 2−d
2 2j 22k
.
(6.5)
The first estimate is trivial and the second follows from Lemma 3.1 as follows. The curve
γσk−1 is a crosscut of D = 2k−1 D∩H and so partitions D into exactly two components, one
of which contains z. Consequently we get an upper bound on Sσk−1 (z) by estimating the
probability of a Brownian motion from z to reach distance 2k−1 from 0 before hitting the
real line or the curve. Thus, given the path up to time σk−1 , the Beurling estimate shows
that the probability that a Brownian motion starting at z reaches the circle of radius
2 Im z about z without exiting Hσk−1 is O((2j / Im z)1/2 ). Given this, the gambler’s ruin
estimate shows that the probability to reach modulus 2k−1 is O(Im z/2k ). Hence, since
Im z 6 1, we see that Sσβk−1 (z) 6 c (2j /22k )β/2 . This gives (6.5). By Lemma 3.1 we have
P Vjk ∩ {σk−1 < ∞} 6 C(2j−1 )2−d S0β ,
which combined with (6.5) gives
b 21 log2 −1 c
P τ < ∞, E c , Uk , Vjk 6 C 2−d+β/4 2−kβ .
X
(6.6)
j=−1
It remains to handle the terms with j > 21 log2 −1 so that 2j > 21/2 which we now
assume. Lemma 6.1 implies that there is α > 0 such that on the event {τ01/2 < ∞},
P τ < ∞, E γτ 0
6 C (2−d)/2+α .
1/2
c
Moreover, on the event Vjk ∩{σk−1 < ∞}, we can use Lemma 3.1 and the Beurling estimate
as above to see that
P
τ01/2
< ∞ | γσk−1 6 C
1/2
2j !2−d
·
2j 22k
!β/2
We conclude that
P τ < ∞, E c , Uk , Vjk 6 C 2−d+α (2j )β/2 2−βk .
32
.
So summing this over j = d 12 log2 −1 e, . . . , J and using also (6.6) shows that
2kr P (τ < ∞, E c , Uk ) 6 2k(r−β) o(2−d ).
Since r − β < 0 (equivalent to the condition ρ < 8 − κ) this is summable over k and the
result is o(2−d ). This proves (6.2).
Given (6.2) it is enough to prove that
lim d−2 Pρ (τ < ∞, E) = c∗ Gκ,ρ (z, ξ 1 , ξ 2 ).
↓0
For this let us fix > 0 for the moment and recall that we write τ = τ . According to
equation (3.14), we have
E∗ [f ] = G̃−1
t > 0,
0 E[G̃t f ],
whenever f ∈ L1 (P∗ ) is measurable with respect to F̃t . We change to the radial time
1
ln , so that = e−2at and s(t) = τ . Then G̃t = Gτ and
parmetrization and set t = − 2a
(ρ)
the function Mτ 1τ <∞ 1E is measurable with respect to F̃t = Fτ , so we find
h
i
h
i
(ρ)
Pρ (τ < ∞, E) = E Mτ(ρ) 1τ <∞ 1E = G0 E∗ G−1
τ Mτ 1τ <∞ 1E ,
(6.7)
where G0 is the SLEκ Green’s function. Thanks to the boundary conditions of the
martingale Gt , we have Gs(t) = G∞ = 0 on the event τ = ∞. This means that
E∗ [1τ =∞ ] = G−1
0 E[G∞ 1τ =∞ ] = 0. Hence we can remove the factor 1τ <∞ from the
right-hand side of (6.7). Thus, using the definition (3.12) of G,
h
i
Pρ (τ < ∞, E) = 2−d G0 E∗ Mτ(ρ) Sτ−β 1E ,
where Sτ = Sτ (z). We need to show that
h
i
h
(ρ)
lim E∗ Mτ(ρ) Sτ−β 1E = c∗ E∗ MT
↓0
i
,
τ = τ ,
(6.8)
and where T is the time at which the path reaches z. Let τ 00 = τ1/2 /4 . Then τ01/2 6 τ 00 6 τ
if is small enough. The Beurling estimate implies that
(ρ) (ρ)
Mτ − Mτ 00 1E = O(r/8 ).
h
i
Using the invariant distribution (see Lemma 3.4) we have that E∗ Sτ−β = O(1), so
h
i
h
i
(ρ)
∗
Mτ(ρ) − Mτ 00 Sτ−β 1E 6 Cr/8 E∗ Sτ−β = O(r/8 ),
E
(ρ)
On the other hand, since Mτ 00 1Uk 1τ <∞ 6 C2kr the same argument that proved (6.2)
shows that
h
i
(ρ)
E∗ Mτ 00 Sτ−β 1E c = o(1)
as → 0. In other words,
h
i
h
(ρ)
i
E∗ Mτ(ρ) Sτ−β 1E = E∗ Mτ 00 Sτ−β + o(1).
33
Moreover,
h
h
i
(ρ)
h
(ρ)
E∗ Mτ 00 Sτ−β = E∗ Mτ 00 E∗ Sτ−β | Fτ 00
ii
.
Using Lemma 3.4 we see that there is α > 0 such that
i
h
E∗ Sτ−β | Fτ 00 =
c∗
2
Z π
sin θ dθ (1 + O(α )) = c∗ (1 + O(α )).
0
It only remains to show that
h
h
i
(ρ)
lim E∗ Mτ(ρ) = E∗ MT
↓0
i
.
(ρ)
For this we need to check that the sequence of integrands (Mτ ) is uniformly integrable
(ρ)
as ↓ 0, that is, that for each 0 > 0 there exists R < ∞ such that E∗ [Mτ 1M (ρ) >R ] < 0
τ
(ρ)
uniformly in . Since the only way in which Mτ can get large is by the path reaching a
large diameter, this follows from a similar (but easier) argument as the one proving (6.2).
We omit the details, but remark that this argument also needs r − β < 0. The Vitali
(ρ)
(ρ)
convergence theorem now implies that Mτ converges to MT in L1 (P∗ ). The proof is
complete.
6.2
Probabilistic representation for GCFT : Proof of Proposition 2.13
Let 0 < κ 6 4. Our goal is to show that
(2)
G(z, ξ 1 , ξ 2 ) = (Im z)d−2 sinβ (arg(z − ξ 1 ))E∗ [MT ],
z ∈ H,
ξ1 < ξ2,
(6.9)
where E∗ denotes expectation with respect to two-sided radial SLEκ from ξ 1 through z,
stopped at the hitting time T of z. Our first step is to use scale and translation invariance
to reduce the relation (6.9), which depends on the four real variables x = Re z, y =
Im z, ξ 1 , ξ 2 , to an equation involving only two independent variables.
6.2.1
The function h(θ1 , θ2 )
It follows from (2.7) and (2.8) that G satisfies the scaling behavior
G(λz, λξ 1 , λξ 2 ) = λd−2 G(z, ξ 1 , ξ 2 ),
λ > 0.
Hence we can write
G(z, ξ 1 , ξ 2 ) = y d−2 H(z, ξ 1 , ξ 2 ),
where the function H is homogeneous and translation invariant:
H(λz, λξ 1 , λξ 2 ) = H(z, ξ 1 , ξ 2 ),
1
2
1
λ > 0,
2
H(z, ξ , ξ ) = H(x + λ, y, ξ + λ, ξ + λ),
(6.10a)
λ ∈ R.
(6.10b)
It follows that the value of H(x, y, ξ 1 , ξ 2 ) only depends on the two angles θ1 and θ2 defined
by
θ1 = arg(z − ξ 1 ),
θ2 = arg(z − ξ 2 ).
34
Figure 5. The graph of the function h(θ1 , θ2 ) for α = 5/2.
In particular, if we let ∆ denote the triangular domain
∆ = {(θ1 , θ2 ) ∈ R2 | 0 < θ1 < θ2 < π},
then we can define a function h : ∆ → R for α ∈ (1, ∞) \ Z by the equation
G(z, ξ 1 , ξ 2 ) = y d−2 h(θ1 , θ2 ),
z ∈ H, −∞ < ξ 1 < ξ 2 < ∞.
(6.11)
According to Lemma 9.4, we can extend the definition of h to all α ∈ (1, ∞) by continuity.
We write h(θ1 , θ2 ; α) if we want to indicate the α-dependence of h(θ1 , θ2 ) explicitly. In
terms of h, we can then reformulate equation (6.9) as follows:
(2)
h(θ1 , θ2 ; α) = (sinβ θ1 )E∗ [MT ],
(θ1 , θ2 ) ∈ ∆,
β > 1.
(6.12)
The following lemma, which is crucial for the proof of (6.12), describes the behavior of
h near the boundary of ∆. In particular, it shows that h(θ1 , θ2 ) vanishes as θ1 approaches
0 or π, and that the restriction of h to the top edge θ2 = π of ∆ equals sinβ θ1 . In other
words, the lemma verifies that G(z, ξ 1 , ξ 2 ) satisfies the appropriate boundary conditions.
Lemma 6.2 (Boundary behavior of h). Let α > 2. Then the function h(θ1 , θ2 ) defined in
(6.11) is a smooth function of (θ1 , θ2 ) ∈ ∆ and has a continuous extension to the closure
¯ of ∆. This extension satisfies
∆
h(θ1 , π) = sinβ θ1 ,
θ1 ∈ [0, π],
θ ∈ (0, π),
h(θ, θ) = hf (θ),
(6.13)
(6.14)
where hf (θ) is defined in (8.8). Moreover, there exists a constant C > 0 such that
0 6 h(θ1 , θ2 ) 6 C sinβ θ1 ,
¯
(θ1 , θ2 ) ∈ ∆,
(6.15)
and
|h(θ1 , θ2 ) − h(θ1 , π)|
|π − θ2 |
6
C
,
sin θ1
sinβ θ1
35
(θ1 , θ2 ) ∈ ∆.
(6.16)
Proof. The rather technical proof involves asymptotic estimates of the integral in (2.7)
and is given in Appendix B.
The derivation of formula (6.12) relies on an application of the optional stopping
theorem to the martingale observable associated with G. The following lemma gives an
expression for this local martingale in terms of h.
Lemma 6.3 (Martingale observable for SLEκ (2)). Let θtj = arg(gt (z) − ξtj ), j = 1, 2.
Then
Mt = Υt (z)d−2 h(θt1 , θt2 )
(6.17)
is a local martingale for SLEκ (2) started from (ξ 1 , ξ 2 ).
Proof. The proof follows from a direct computation using Itô’s formula. In fact, since
Υd−2
h(θt1 , θt2 ) = |gt0 (z)|2−d G(Zt , ξt1 , ξt2 ),
t
we see that Mt is the martingale observable relevant for the Green’s function found in
Section 4 (cf. equation (4.14)). Hence, the result is a special case of Lemma 4.2 in the
case when the growth speed λ2 (t) of the second curve vanishes and the parameter σ1 , σ2 , s
are given by (4.12).
Remark 6.4. The observable (6.17) is a local martingale also for the system of two
commuting SLEs started from (ξ 1 , ξ 2 ) by the same proof.
Let z ∈ H and consider SLEκ (2) started from (ξ 1 , ξ 2 ) with ξ 1 < ξ 2 . For each > 0,
we define the stopping time τ by
τ = inf{t > 0 : Υt 6 Υ0 },
where Υt = Υt (z). Let > 0 and n > 1. Then, since Υt is a nonincreasing function of t
and Υ0 = y, we have
y > Υt∧n∧τ > Υτ = y,
t > 0.
(6.18)
Hence, in view of the boundedness (6.15) of h, Lemma 6.3 implies that (Mt∧τ ∧n )t>0 is a
true martingale for SLEκ (2). The optional stopping theorem therefore shows that
1
2
h(θ1 , θ2 ) = Υ02−d E2 Υd−2
n∧τ h(θn∧τ , θn∧τ ) .
(6.19)
Recall that P and P2 denote the SLEκ and SLEκ (2) measures respectively, and that
E and E2 denote expectations with respect to these measures. Equations (3.7) and (3.8)
imply
(2) P2 (V ) = E Mt 1V
for V ∈ Ft ,
where
(2)
Mt
=
ξt2 − ξt1
ξ2 − ξ1
!a
gt0 (ξ 2 )
3a−1
2
.
In particular,
(2)
E2 [f ] = E Mn∧τ f ,
(2)
whenever Mn∧τ f is an Fn∧τ -measurable L1 (P) random variable.
36
(6.20)
(2)
Lemma 6.5. For each t > 0, we have Mt
∈ L1 (P).
Proof. The identity
gt0 (ξ 2 ) = exp
−
Z t
0
ads
2
(ξs − ξs1 )2
shows that
0 6 gt0 (ξ 2 ) 6 1,
t > 0.
(6.21)
Moreover, if Et denotes the interval Et = (ξt1 , ξt2 ) ⊂ R, then conformal invariance of
harmonic measure gives
lim sπω(is, gt−1 (Et ), H \ γt ) = lim sπω(gt (is), Et , H) = |ξt2 − ξt1 |.
s→∞
s→∞
Since the left-hand side is bounded above by a constant times 1 + diam(γt ), this gives the
estimate
|ξt2 − ξt1 | 6 C(1 + diam(γt )),
t > 0.
(6.22)
On the other hand, since ξt1 is the driving function for the Loewner chain gt , we have (see
e.g. Lemma 4.13 in [27])
diam(γt ) 6 C max
n√
o
t, sup |ξs1 | ,
t > 0.
06s6t
Combining the above estimates, we find
(2)
|Mt | 6 C |ξt2 − ξt1 |a 6 C(1 + diam(γt ))a 6 C 1 + max
(2)
Since ξt1 is a P-Brownian motion, it follows that Mt
n√
oa
t, sup |ξs1 |
,
06s6t
t > 0.
∈ L1 (P) for each t > 0.
As a consequence of (6.15), (6.18), and Lemma 6.5, the random variable
(2)
1
2
Mn∧τ Υd−2
n∧τ h(θn∧τ , θn∧τ )
is Fn∧τ -measurable and belongs to L1 (P). Thus, we can use (6.20) to rewrite (6.19) as
(2)
d−2
1
2
h(θ1 , θ2 ) = Υ2−d
0 E Mn∧τ Υn∧τ h(θn∧τ , θn∧τ ) .
We split this into two terms depending on whether τ 6 n or τ > n as follows:
(2) d−2
1
2
1 2
h(θ1 , θ2 ) = Υ2−d
0 E Mτ Υτ h(θτ , θτ )1τ 6n + F,n (θ , θ ),
(6.23)
where
1 2
F,n (θ1 , θ2 ) = Υ02−d E Mn(2) Υd−2
n h(θn , θn )1τ >n .
We prove in Lemma 6.7 below that F,n (θ1 , θ2 ) → 0 as n → ∞ for each fixed > 0.
Assuming this, we conclude from (6.23) that
1
2
h(θ1 , θ2 ) = Υ2−d
lim E Mτ(2)
Υd−2
τ h(θτ , θτ )1τ 6n .
0
n→∞
37
(6.24)
Equations (3.12) and (3.14) imply
P∗ (V ) = G−1
0 E[Gt 1V ] for V ∈ Ft ,
(6.25)
where Gt = Υd−2
sinβ θt1 . In particular,
t
(2)
E Mτ(2)
f = G0 E∗ G−1
τ Mτ f ,
(2)
whenever Mτ f is an Fτ -measurable random variable in L1 (P). Using Lemma 6.5 again,
(2)
1
1
2
we see that the function Mτ Υd−2
τ h(θτ , θτ )1τ 6n is Fτ -measurable and belongs to L (P)
for n > 1 and > 0. Thus (6.24) can be expressed in terms of an expectation for two-sided
radial SLEκ through z as follows:
∗
−1
(2) d−2
1
2
h(θ1 , θ2 ) = Υ2−d
0 G0 lim E Gτ Mτ Υτ h(θτ , θτ )1τ 6n
=
n→∞
2−d
(2) d−2
1
2 Υ0 G0 E∗ G−1
τ Mτ Υτ h(θτ , θτ ) ,
(6.26)
where the second equality is a consequence of dominated convergence and the fact that
E∗ [1τ <∞ ] = 1. Using that Gt = Υtd−2 sinβ θt1 , we arrive at
h(θ1 , θ2 ) = sinβ θ1 E∗ Mτ(2)
h(θτ1 , θτ2 ) .
sinβ θτ1
(6.27)
In the limit as τ → T , where T denotes the hitting time of z, we have θτ2 → θT2 = π.
Hence we use (6.13) to write (6.27) as
+ E(θ1 , θ2 ),
h(θ1 , θ2 ) = sinβ θ1 E∗ Mτ(2)
where
E(θ1 , θ2 ) = sinβ θ1 E∗ Mτ(2)
h(θτ1 , θτ2 ) − h(θτ1 , π)
.
sinβ θτ1
But the estimate (6.16) implies
h
i
|E(θ1 , θ2 )| 6 c E∗ Mτ(2)
|π − θτ2 |(sin θτ1 )−1 .
Equation (6.12) therefore follows from the following lemma.
Lemma 6.6. For any (θ1 , θ2 ) ∈ ∆, it holds that
h
i
h
(2)
lim E∗ Mτ(2)
= E ∗ MT
i
(6.28)
i
(6.29)
↓0
and
h
lim E∗ Mτ(2)
|π − θτ2 |(sin θτ1 )−1 = 0.
↓0
38
(2)
Proof. For (6.28) it is enough to show that the family {Mτ } is uniformly integrable, i.e.,
that for every 0 > 0, there exists an R > 0 such that
h
i
1{M (2) >R} < 0
E∗ Mτ(2)
(6.30)
τ
for all small > 0.
Let us prove (6.30). For R > 0, we define the stopping time λR by
λR = inf{t > 0 : |γ(t)| = R}
and write {τ < ∞} = ∪∞
j=0 Ej (), where
Ej () = {λ2j 6 τ < λ2j+1 }.
The estimate (6.22) yields
|ξτ2 − ξτ1 | 6 c 2j+1
on Ej (),
j > 0,
so, in view of (6.21), there exists a constant c0 > 0 such that
|Mτ(2)
| 6 c0 2(j+1)a
on Ej (),
j > 0.
Here and below the constants C and c0 are independent of > 0, j > 0, and R > 0.
Suppose R > 0 and let N := [a−1 log2 (R/c0 )] denote the integer part of a−1 log2 (R/c0 ).
Then
|Mτ(2)
| 6 c0 2(j+1)a 6 R on Ej (),
0 6 j 6 N − 1.
Hence
E∗ [Mτ(2)
1{M (2) >R} ] 6 E∗ [Mτ(2)
1∪∞
] 6 c0
j=N Ej ()
τ
∞
X
2(j+1)a E∗ [Ej ()].
(6.31)
j=N
The set Ej () is Fτ -measurable, so equation (6.25) gives
d−2
P∗ (Ej ()) = P∗ (Ej ()1τ <∞ ) = G−1
E[1Ej () 1τ <∞ ],
0 E[Gτ 1Ej () 1τ <∞ ] 6 C
where we have used the following estimate in the last step:
|Gτ | = (Υ0 )d−2 sinβ θτ1 6 Cd−2 .
We claim that
P (Ej () τ < ∞) 6 C2−d 2−jβ
for j > log2 (4|z|).
Assuming for the moment that (6.32) holds, we find
P∗ (Ej ()) 6 C2−jβ ,
j > log2 (4|z|).
Employing this estimate in (6.31) we obtain
E∗ [Mτ(2)
1{M (2) >R} ] 6 C
τ
∞
X
2(j+1)a 2−jβ 6 C
j=N
∞
X
j=N
39
2−j(3a−1)
(6.32)
6 C2
−N (3a−1)
R
6C
c0
3a−1
a
,
> 0,
N > log2 (4|z|).
The condition N > log2 (4|z|) is fulfilled for all sufficiently large R. Hence, given 0 > 0, by
(2)
choosing R large enough, we can make E∗ [Mτ 1{M (2) >R} ] < 0 for all > 0. This proves
τ
(6.30), assuming (6.32) which we now verify. Suppose j > log2 (4|z|), i.e., |z| < 2j /4. Let
Dj = H \ γ([0, λ2j ]) and let gj : Dj → H be the uniformizing map with gj (γ(λ2j )) = 0. Let
k > 1 be the unique integer such that 2yk < Υλ2j 6 22yk . By (3.2), sin arg(gj (z)) is bounded
above by a constant times the probability that a Brownian motion starting at z reaches
the circle of radius 2j centered at the origin without leaving Dj . By a Beurling estimate,
the probability that it reaches the circle of radius 2y centered at the origin without leaving
Dj is bounded by C2−k/2 , and given this the probability to reach the circle of radius 2j
is bounded by Cy2−j . Hence
1/2 √
sin arg(gj (z)) 6 C2−k/2 y2−j 6 CΥλ
2j
y2−j .
(6.33)
On the other hand, by Lemma 3.1,
Υ0
P (Ej ()) 6 C
Υλ2j
2−d
sinβ (arg gj (z)).
(6.34)
Combining (6.33) and (6.34), we find
2−d
P (Ej ()) 6 C
Υ0
Υλ2j
2−d
β
d−2+ β2
β
Υλ2 j y 2 2−jβ 6 C2−d 2−jβ Υλ
2
2j
.
Since d − 2 + β/2 > 0 for 0 < κ 6 4, this proves (6.32). This completes the proof of (6.28).
It remains to prove (6.29). Note that there is a constant c (depending on z) such that
h
i
h
i
E∗ Mτ(2)
|π − θτ2 |(sin θτ1 )−1 6 c 1/2 E∗ Mτ(2)
(sin θτ1 )−1 .
Indeed, |π − θτ2 | is bounded above by a constant times the harmonic measure from z in
Hτ of [ξ 2 , ∞), which by the Beurling estimate is O(1/2 ). On the other hand, recalling
the definition of the measure P∗ and that β − 1 > 0 when κ 6 4, we see that
h
i
E∗ Mτ(2)
(sin θτ1 )−1 =
h
i
h
i
d−2
d−2
(2) β−1
(2)
E
M
S
1
6
E
M
1
.
τ
<∞
τ
<∞
τ
τ
τ
sinβ θ1
sinβ θ1
Using Proposition 2.12 we see that last term converges, and is in particular bounded as
→ 0. This completes the proof, assuming Lemma 6.7.
Lemma 6.7. Let
1 2
F,n (θ1 , θ2 ) = Υ02−d E Mn(2) Υd−2
n h(θn , θn )1τ >n .
For each > 0,
lim F,n (θ1 , θ2 ) = 0.
n→∞
40
Proof. We can without loss in generality assume that |z| 6 1. All constants are allowed
to depend on z. Recall that
(2)
Mt
We know that |gt0 (ξ 2 )|
enough to prove that
3a−1
2
ξt2 − ξt1
ξ2 − ξ1
=
!a
gt0 (ξ 2 )
3a−1
2
.
6 1, Υnd−2 1τ >n 6 Cd−2 , and h(θn1 , θn2 ) 6 C sinβ θn1 , so it is
h
i
lim E |ξn2 − ξn1 |a sinβ θn1 = 0.
n→∞
√
√
For k = 1, 2, . . ., let Uk be the event that 2k 2an 6 rad γ[0, n] 6 2k+1 2an, where
rad K = sup{|z| : z ∈ K}. Since we have parametrized so that hcap γ[0, t] = at, we have
P(∪∞
k>0 Uk ) = 1. Fix 0 < p < 1/4. For each integer j > 0, let Vj be the event that
2j np 6 |γ(n)| 6 2j+1 np and let V−1 be the event that |γ(n)| < np . (We could phrase these
events in terms of stopping times.) We write
h
i
E |ξn2 − ξn1 |a sinβ θn1 6
X
h
E |ξn2 − ξn1 |a sinβ θn1 1Uk
i
k>0
For each k we will estimate
h
i
E |ξn2 − ξn1 |a sinβ θn1 1Uk 6
J
X
h
i
E |ξn2 − ξn1 |a sinβ θn1 1Vj 1Uk ,
j=−1
where J = d( 21 − p) log2 (n) + k + 12 log2 (2a)e. By Theorem 1.1 of [18] we have for j =
−1, 0, 1, . . .,
P(Uk ∩ Vj ) 6 C(np−1/2 2j−k )β P(Uk ) 6 C(np−1/2 2j−k )β .
(6.35)
On the event V−1 ∩ Uk we then use the trivial estimate sin θn1 6 1 to find
h
i
E |ξn2 − ξn1 |a sinβ θn1 1V−1 1Uk 6 C(2k n1/2 )a P(Uk ∩ V−1 ) 6 C(2k n1/2 )a (np−1/2 2−k )β .
Since a − β < 0 for a > 1/3 this is summable over k, and we see that
h
i
E |ξn2 − ξn1 |a sinβ θn1 1V−1 6 Cn−((1/2−p)β−a/2) .
(6.36)
When j > 0, we can estimate using the Beurling and Gambler’s ruin estimates (see the
previous lemma), to find
sinβ θn1 1Vj 6 C(n−p 2−j )β 1Vj ,
and therefore
|ξn2 − ξn1 |a sinβ θn1 1Vj 1Uk 6 C(n1/2 2k )a (n−p 2−j )β 1Vj 1Uk .
So using (6.35),
J
X
h
i
E |ξn2 − ξn1 |a sinβ θn1 1Vj 1Uk 6 Cp 2−k(β−a) n−(β/2−a/2) (log2 n + k).
j=0
This is summable over k when a > 1/3 for any choice of p and the sum is o(1) as n → ∞.
Since 0 < p < 1/4, the exponent in (6.36) is strictly negative whenever a > 1/2. The
proof is complete.
41
7
Two paths getting near the same point: Proof of Lemma
2.9
This section proves the correlation estimate Lemma 2.9 and this will complete the proof
of Theorem 2.6.
Lemma 7.1. For any ∈ (0, 1/2), it holds that
√
8 ∂n ω − 1, ∂B (0), D \ (B (0) ∪ [0, 1]) 6
.
π
√
√
Proof. Assume first that ∈ (0, 1). Let z1 = −1/ z, where the branch cut for z runs
along R+ . The map z 7→ z1 takes Ω := D \ (B (0) ∪ [0, 1]) onto Ω1 := {ρeiϕ : ρ ∈
(1, −1/2 ), ϕ ∈ (0, π)}. The Joukowski transform z1 7→ z2 := (z1 + z1−1 )/2 then maps Ω1
conformally onto the semi-ellipse Ω2 := E ∩ H, where
x2
o
y2
E = x + iy 2 + 2 < 1 ,
r
r
n
+
−
1
1
− 2 ± 2
r± =
.
2
The composed map f defined by f (z) = z2 is a conformal map Ω → Ω2 such that
f (−1) = 0 and f (∂B (0)) = H ∩ ∂E. Hence
∂n ω(−1, ∂B (0), Ω) = |f 0 (−1)|∂n ω(0, H ∩ ∂E, Ω2 ).
For z ∈ H ∩ Br− (0), we have the estimate
ω(z, H ∩ ∂E, Ω2 ) 6 ω(z, H ∩ ∂Br− (0), H ∩ Br− (0)) =
z − r 2
−
π − arg
,
π
z + r−
where the explicit expression for the harmonic measure on the right-hand side can be
−
verified by noting that arg( z−r
z+r− ) is a harmonic function of z in the semidisk H ∩ Br− (0)
with boundary values π/2 and π on H ∩ ∂Br− (0) and [−r− , r− ], respectively. Using that
f 0 (−1) = i/2, this gives
∂n ω(−1, ∂B (0), Ω) 6 |f 0 (−1)|∂n ω(0, H ∩ ∂Br− (0), H ∩ Br− (0))
iy − r 1 ∂ 2
−
=
π − Im ln
=
,
π ∂y y=0
iy + r−
πr−
√
Since πr2− 6 8 /π for ∈ (0, 1/2), the lemma follows.
∈ (0, 1).
Lemma 7.2. Let z = x + iy ∈ H. Let > 0 be given with < y/10 and let B := B (z).
Let γ be a simple curve connecting 0 with the closed disc B̄ and staying in H \ B̄ except
for the endpoints. Then
EH\(γ∪B̄) (R+ , ∂B) 6 C(/y)1/2 ,
where the constant C < ∞ is independent of γ, z ∈ H, and ∈ (0, y/10).
42
Proof. Let ϕ : H → D be the conformal map with ϕ(z) = 0 and ϕ(0) = 1. Note that
|ϕ0 (z)| = 1/(2y). Write γ 0 , B 0 for the images of γ, B under ϕ and let D0 = D \ (γ 0 ∪ B 0 ).
By the distortion estimate (3.1), there is a c > 0 such that B 0 ⊂ B 00 := {z : |z| < c(/y)}.
Monotonicity of harmonic measure and Beurling’s projection theorem then imply that,
for all ζ ∈ ∂D \ {1},
∂n ω(ζ, ∂B 0 ; D0 ) 6 ∂n ω(−1, ∂B 00 ; D \ (B 00 ∪ [0, 1])).
But Lemma 7.1 shows that
00
∂n ω −1, ∂B ; D \
(B 00
p
8 c/y
∪ [0, 1]) 6
.
π
Hence
EH\(γ∪B̄) (R+ , ∂B) 6 ED0 (∂D \ {1}, ∂B 0 ) . (/y)1/2 .
We could quote Theorem 1.8 of [32] for a slightly different version of the next lemma,
but since our proof is short and also that slightly different we will give it here.
Lemma 7.3. Suppose κ > 0 and ρ > max{−2, κ/2 − 4} and consider SLEκ (ρ) started
from (0, 1). Let C∞ denote the function Ct defined in (3.10) evaluated at t = ∞. Then
there exists a q > 0 such that
Pρ0,1 (C∞ (1) 6 ) = c̃ β+ρa (1 + O(q )) ,
↓ 0,
(7.1)
where the constant c̃ = c̃(κ, ρ) is given by
c̃ =
Γ(6a + aρ)
.
2aΓ(2a)Γ(4a + aρ)
In particular, there is a constant C < ∞ such that
Pρ0,1 (γ ∩ η 6= ∅) 6 C EH\η (R− , η)β+ρa
for all crosscuts η separating 1 from 0 in H.
Proof. Write ξt1 for the driving term of γ and let ξt2 = gt (1), where gt is the Loewner chain
of γ. We get SLEκ (ρ) started from (0, 1) by weighting SLEκ by the local martingale (see
(3.7))
(ρ)
Mt = (ξt2 − ξt1 )r gt0 (1)ζ(r) , r = ρa/2.
Let
Nt = Ct (1)−(β+aρ) Aβ+aρ
,
t
At =
ξt2 − Ot
.
ξt2 − ξt1
Direct computation shows that Nt is a local martingale for SLEκ (ρ) started from (0, 1),
which satisfies N0 = 1. Moreover,
(ρ)
(κ−8−ρ)
Mt Nt = Mt
43
,
(κ−8−ρ)
where Mt
is the local SLEκ martingale corresponding to the choice r = rκ (κ − 8 −
ρ) = −β − ρa/2. We will work in the radial parametrization seen from 1. We set
s(t) = inf{s > 0 : Cs (1) = e−at }
(ρ)
and write M̂tρ = Ms(t) , etc. for the time-changed processes. We have
h
Pρ0,1 (s(t) < ∞) = E M̂tρ 1s(t)<∞
i
h
= E M̂tρ N̂t N̂t−1 1s(t)<∞
i
−(β+aρ)
h
= e−a(β+aρ) E M̂tκ−8−ρ Ât
h
−(β+aρ)
= e−a(β+aρ) E∗ Ât
1s(t)<∞
i
i
1s(t)<∞ ,
where E∗ refers to expectation with respect to SLEκ (κ − 8 − ρ) started from (0, 1). The
exponent is positive if ρ > κ/2 − 4 and ρ̃ = κ − 8 − ρ < κ/2 − 4. The key observation is
that under the measure P∗ we have that s(t) < ∞ almost surely and that Ât is positive
recurrent and converges to an invariant distribution. This uses ρ > κ/2 − 4; see e.g.,
[1, 28]. In fact, we have the following formula for the limiting distribution (set ν =
−rκ (κ − 8 − ρ) = β + aρ/2 in Lemma 2.2 of [1]):
π(x) = c0 xβ+aρ (1 − x)2a−1 ,
c0 =
Γ(6a + aρ)
;
Γ(2a)Γ(4a + aρ)
It follows that
h
−(β+aρ)
E∗ Ât
i
1s(t)<∞ = c0
Z 1
(1 − x)2a−1 dx 1 + O(e−qt ) ,
0
which gives (7.1). By the distortion estimates (3.1), if τ = inf{t > 0 : dist(γt , 1) 6 ),
then
Pρ0,1 (τ < ∞) β+aρ .
The last assertion then follows using
EH\η (R− , η) &
diam(η)
∧ 1.
dist(0, η)
Lemma 7.4. There is a constant 0 < c < ∞ such that the following holds. Let D
be a simply connected domain containing 0 and with three marked boundary points ζ, ξ, η.
Suppose γζ , γξ are crosscuts of D which are disjoint except at η, and which connects η with
ζ and η with ξ, respectively, and such that neither crosscut disconnects 0 from the other.
Write Dζ and Dξ for the components of 0 of D \ γζ and D \ γξ and let Dζ,ξ = Dζ ∩ Dξ .
Suppose 0 < < r/10 and
rDζ (0) 6 4, rDξ (0) > r.
(7.2)
Then if
rDζ (0) > (1 + 3c/r) ,
it holds that
rDζ,ξ (0) > (1 + c/r) .
44
Proof. Let φζ : Dζ → D be the conformal map with φζ (0) = 0, φ0ζ (0) > 0. Note that
φζ (γξ ) is a crosscut of D. By distortion estimates, dist(0, ∂Dζ ) rDζ (0) while dist(0, γξ ) >
rDξ (0)/4. Therefore the Beurling estimate and the bounds in (7.2) show that there is a
universal constant c1 such that
diam φζ (γξ ) 6 c1 (/r)1/2 .
(7.3)
Write D0 for the component containing 0 of D r φζ (γξ ). Note that 0 ∈ D0 and let
ψ : D0 → D with ψ(0) = 0, ψ 0 (0) > 0. Then the normalized Riemann map of Dζ,ξ is ψ ◦ φζ
and so we have
rDζ,ξ (0) = rDζ (0)ψ 0 (0)−1 .
By (7.3), we have diam(D \ D0 ) 6 c2 (/r)1/2 , so the logarithmic capacity of D \ D0 is at
most a universal constant c3 times /r. Therefore,
1 6 ψ 0 (0) 6 1 + c4 /r.
Hence
rDζ,ξ (0) > rDζ (0)
1
.
1 + c4 /r
Consequently, if rDζ (0) > (1 + 3c4 /r) we have
rDζ,ξ (0) > (1 + c4 /r) .
Lemma 7.5. There exists a constant C < ∞ such that if 0 < < δ < Im z/10 and
ξ 1 < ξ 2 , then
Pξ1 ,ξ2 Υ1∞ (z) 6 , Υ2∞ (z) 6 6 C (/y)2−d+β/2+a ,
(7.4)
where 2 − d = 1 − 1/4a, β = 4a − 1.
Remark. Notice that if a > 1/2, i.e., κ 6 4, then β/2 > 2 − d, i.e., half the boundary
exponent is larger than the bulk exponent, with strict inequality if κ < 4. Therefore, (7.4)
implies that there for every κ 6 4 is u > 0 such that
Pξ1 ,ξ2 Υ1∞ (z) 6
√
, Υ2∞ (z) 6
√ = Oy (2−d+u ).
(7.5)
Proof of Lemma 7.5. We first grow γ 1 starting from ξ 1 . The distribution is that of an
SLEκ (2) started from (ξ1 , ξ2 ). Let τ be the first time γ 1 hits the ball B = B(z, 4). By
Proposition 2.12 we have
P (τ < ∞) . (/y)2−d .
(7.6)
On the event that τ < ∞, we stop γ 1 at τ and write H1 = H \ γτ1 . Then almost surely,
C := ∂B ∩ Hτ1 is a crosscut of H1 . By Lemma 7.2 we have
EH1 \C ([ξ2 , ∞), C ) . (/y)1/2 ,
45
(7.7)
on the event that τ < ∞, with a universal constant. Conditioned on γτ1 (after uniformizing H1 ) the distribution of γ 2 is that of SLEκ (2) started from (ξτ2 , ξτ1 ). We claim that on
the event that τ < ∞
P γ 2 ∩ C 6= ∅ | γτ1 . (/y)β/2+a .
(7.8)
Indeed, note that gτ (C ) is a crosscut of H separating ξτ1 from ξτ2 and by (7.7) and
conformal invariance
EH\gτ (C ) ([ξτ2 , ∞), gτ (C )) . (/y)1/2 .
The estimate (7.8) now follows from Lemma 7.3 with ρ = 2. We conclude the proof by
combining (7.8) with (7.6).
We can now give the proof of Lemma 2.9.
Proof of Lemma 2.9. Consider a system of commuting SLEs started from ξ 1 < ξ 2 . Then
we want to prove that
lim d−2 P (Υ∞ (z) 6 ) = lim d−2 P Υ1∞ (z) 6 + lim d−2 P Υ2∞ (z) 6 .
↓0
↓0
↓0
We can write
P (Υ∞ (z) 6 ) = P Υ1∞ (z) 6 + P Υ2∞ (z) 6 − P Υ1∞ (z) 6 , Υ2∞ (z) 6 + P Υ1∞ (z) > , Υ2∞ (z) > , Υ∞ (z) 6 .
We know from Theorem 2.12 that the renormalized limits of the first two terms on the
right exist. We will show that the remaining terms decay as o(2−d ) and this will prove the
lemma. For the third term the required estimate follows immediately from Lemma 7.5,
so it remains to estimate the last term.
By distortion estimates we have that Υ∞ 6 implies dist(γ 1 ∪ γ 2 , z) 6 2. We may
assume that dist(γ 1 , z) 6 2, which in turn implies Υ1∞ (z) 6 4. Using Lemma 7.4 with
√
r = we see that there is a constant c such that the following estimates hold:
P(Υ1∞ (z) > , Υ2∞ (z) > , Υ∞ (z) 6 )
√
√ 6 P (1 + c ) 6 Υ1∞ (z) 6 4, Υ2∞ (z) 6 √ + P 6 Υ1∞ (z) 6 (1 + c )
√
√ 6 P Υ1∞ (z) 6 , Υ2∞ (z) 6 √ + P 6 Υ1∞ (z) 6 (1 + c ) .
By (7.5) and the fact that β/2 + a > 2 − d,
P Υ1∞ (z) 6
√
, Υ2∞ (z) 6
√ = O((2−d+β/2+a)/2 ) = o(2−d ).
46
We can use Theorem 2.12 to see that
h
i
√
√ P 6 Υ1∞ (z) 6 (1 + c ) = c∗ G(2) (z, ξ 1 , ξ 2 )2−d (1 + c )2−d − 1 + o(1)
= o(2−d ).
(Again the error term depends on z and a.) This completes the proof.
8
8.1
Fusion
Schramm’s formula
The function P (z, ξ) in (2.5) extends continuously to ξ = 0; hence we obtain an expression
for Schramm’s formula in the fusion limit by simply setting ξ = 0 in the formulas of
Theorem 2.1. In this way, we recover the formula of [20] and can give a rigorous proof of
this formula.
Theorem 8.1 (Schramm’s formula for fused SLEκ (2)). Let 0 < κ < 8. Consider chordal
SLEκ (2) started from (0, 0+). Then the probability Pf (z) that a given point z = x+iy ∈ H
lies to the left of the curve is given by
1
Pf (z) =
cα
Z ∞
x
Re Mf (x0 + iy)dx0 ,
(8.1)
where cα ∈ R is the normalization constant in (2.6) and
Mf (z) = y α−2 z −α z̄ 2−α
Z z
(u − z)α (u − z̄)α−2 u−α du,
z̄
z ∈ H,
with the contour passing to the right of the origin. The function Pf (z) can be alternatively
expressed as
Γ( α )Γ(α)
Pf (z) = 2−α 2 3α
2
πΓ( 2 − 1)
Z ∞
x
y
S(t0 )dt0 ,
(8.2)
where the real-valued function S(t) is defined by
α 1
1 α
+ , 1 − , ; −t2
2
2
2 2
α
α
2Γ(1 + 2 )Γ( 2 )t
α 3 α 3
2
− 1 α
− , ; −t
,
2 F1 1 + ,
2 2
2 2
Γ( 2 + 2 )Γ(− 21 + α2 )
S(t) = (1 + t2 )1−α
2 F1
t ∈ R.
Remark 8.2. Formula (8.2) for Pf (z) coincides with equation (15) of [20].
Proof of Theorem 8.1. The expression (8.1) for Pf (z) follows immediately by letting ξ → 0
in (2.5). Since the right-hand side of (8.2) vanishes as x → ∞, the representation (8.2)
will follow if we can prove that
Γ( α2 )Γ(α)
y
S(x/y) = Re M(x + iy, 0),
3α
2−α
cα
2
πΓ( 2 − 1)
47
x ∈ R, y > 0, α > 1.
(8.3)
In order to prove (8.3), we write Mf = y α−2 z −α z̄ 2−α Jf (z), where Jf (z) denotes the
function J(z, ξ) defined in (5.2) evaluated at ξ = 0, that is,
Z z
Jf (z) =
(u − z)α (u − z̄)α−2 u−α du,
(8.4)
z̄
where the contour passes to the right of the origin. Let us first assume that x > 0. Then
we can choose the vertical segment [z̄, z] as contour in (8.4). The change of variables
v = u−z
z̄−z , which maps the segment [z, z̄] to the interval [0, 1], yields
Jf (z) = e−iπα (z − z̄)2α−1 z −α
Z 1
v α (1 − v)α−2 1 − v
0
z − z̄ −α
dv,
z
x > 0,
−α for v ∈ [0, 1] and x > 0.
where we have used that (z − v(z − z̄))−α = z −α (1 − v z−z̄
z )
The hypergeometric function 2 F1 can be defined for w ∈ C \ [0, ∞) and 0 < b < c by1
2 F1 (a, b, c; w)
=
Γ(c)
Γ(b)Γ(c − b)
Z 1
v b−1 (1 − wv)−a (1 − v)c−b−1 dv.
0
This gives, for x > 0,
Mf (z) = −iy 3α−3 z −2α z̄ 2−α 22α−1
Γ(α + 1)Γ(α − 1)
z̄ .
2 F1 α, α + 1, 2α; 1 −
Γ(2α)
z
(8.5)
The argument w = 1− z̄z of 2 F1 in (8.5) crosses the branch cut [1, ∞) for x = 0. Therefore,
to extend the formula to x 6 0, we need to find the analytic continuation of 2 F1 . This
can be achieved as follows. Using the general identities
2 F1 (a, b, c; w)
= 2 F1 (b, a, c; w)
and (see [33, Eq. 15.8.13])
2 F1 (a, b, 2b; w)
= 1−
a a + 1
w −a
1 w 2 ,
,b + ;
,
2 F1
2
2
2
2 2−w
we can write the hypergeometric function in (8.5) as
2 F1
α, α + 1, 2α; 1 −
α + 1 α
z̄ x −α−1
1
=
, + 1, α + ; −t−2 ,
2 F1
z
z
2
2
2
x > 0, (8.6)
where t = x/y. Using the identity (see [33, Eq. 15.8.2])
sin(π(b − a)) 2 F1 (a, b, c; w)
(−w)−a 2 F1 (a, a − c + 1, a − b + 1; w1 )
=
π
Γ(c)
Γ(b)Γ(c − a)
Γ(a − b + 1)
1
−b
(−w)
2 F1 (b, b − c + 1, b − a + 1; w )
−
,
w ∈ C \ [0, ∞),
Γ(a)Γ(c − b)
Γ(b − a + 1)
1
Throughout the paper, we use the principal branch of 2 F1 which is defined and analytic for w ∈
C \ [1, ∞).
48
hf (θ)
1.2
1.0
0.8
0.2
0.5
1.0
5.5
8.5
α=
α
=
0.4
α=
2.5
0.6
1.5
2.0
2.5
3.0
θ
Figure 6. The graph of hf (θ) for three different values of α = 8/κ.
with w = −t−2 to rewrite the right-hand side of (8.6), and substituting the resulting
expression into (8.5), we find after simplification
Mf (z) = −
√
i π2α−1 z̄Γ α−1
2
y2Γ
α
2
S(t).
(8.7)
We have derived (8.7) under the assumption that x > 0, but since the hypergeometric
functions in the definition of S(t) are evaluated at the point −t2 which avoids the branch
cut for z ∈ H, equation (8.7) is valid also for x 6 0. Equation (8.3) is the real part of
(8.7).
We obtain Schramm’s formula for commuting SLE in the fusion limit as a corollary.
Corollary 8.3 (Schramm’s formula for two fused commuting SLEs). Let 0 < κ < 8.
Consider two fused commuting SLEκ paths in H started from 0 and growing toward infinity.
Then the probability Pf (z) that a given point z = x + iy ∈ H lies to the left of both curves
is given by (8.1).
Remark 8.4. We remark that the method adopted in [20] was based on exploiting socalled fusion rules, which produces a third order ODE for Pf which can then be solved
in order to give the prediction in (8.2). However, even given the prediction (8.2) for Pf
it is not clear how to proceed to give a proof that it is correct. As soon as the evolution
starts, the tips of the curves are separated and the system leaves the fused state, so it
seems difficult to apply a stopping time argument in this case.
8.2
Green’s function
In this subsection, we derive an expression for the Green’s function for SLEκ (2) started
from (0, 0+). Let α = 8/κ. For α ∈ (1, ∞) \ Z, we define the ‘fused’ function hf (θ) :=
49
i
−
1 + i
L4A
L1A
A
0
L2A
−i
LjA ,
Figure 7. The contours
1
L3A
1+
Re v
1 − i
j = 1, . . . , 4.
hf (θ; α) for 0 < θ < π by
1
π2α+1
1
πα
sin2α−2 (θ) Re e− 2 iπα 2 F1 1 − α, α, 1; (1 − i cot(θ))
sin
ĉ
2
2
hf (θ) =
, (8.8)
where the constant ĉ = ĉ(κ) is defined in (2.9). This definition is motivated by Lemma 6.2,
which shows that hf (θ) is the limiting value of h(θ1 , θ2 ) in the fusion limit (θ1 , θ2 ) → (θ, θ).
The next lemma shows that this definition of hf can be extended by continuity to all α > 1.
Given A ∈ [0, 1] and > 0 small, we let LjA := LjA (), j = 1, . . . , 4, denote the contours
L1A = [A, i] ∪ [i, −],
L2A = [−, −i] ∪ [−i, A],
L3A = [A, 1 − i] ∪ [1 − i, 1 + ],
L4A = [1 + , 1 + i] ∪ [1 + i, A],
oriented so that
P4 j
1 LA
(8.9)
is a counterclockwise contour enclosing 0 and 1, see Figure 7.
Lemma 8.5. For each θ ∈ (0, π), the function hf (θ; α) defined in (8.8) extends to a
continuous function of α ∈ (1, ∞) satisfying
(
n−3
hf (θ; n) = 2
hn sin
Re 2Y2 − iπY1 ,
θ × 2 π Re 2iY2 + πY1 ,
2n−2 1
n = 2, 4, . . . ,
n = 3, 5, . . . ,
(8.10)
where the constant hn ∈ C is defined in (9.8) and the coefficients Yj := Yj (θ; n), j = 1, 2,
are defined as follows: Introduce yj := yj (v, θ; n), j = 0, 1, by
y0 = v n−1 (1 − vz)n−1 (1 − v)−n ,
y1 = v n−1 (1 − vz)n−1 (1 − v)−n ln v + ln(1 − vz) − ln(1 − v) ,
where z =
1−i cot θ
.
2
Then
Y1 = (2πi)2 Res y0 (v, θ; n)
(8.11)
v=1
and
Z
Y2 = 2πi
L1A +L2A +L3A +L4A
y1 dv + 2π
2
Z
L1A −L2A −L3A +L4A
y0 dv,
(8.12)
where 1/z lies exterior to the contours and the principal branch is used for the complex
powers throughout all integrations.
50
Proof. Let n > 2 be an integer. The standard hypergeometric function 2 F1 is defined by
(see [33, Eq. 15.6.5])
2 F1 (a, b, c; z)
e−cπi Γ(c)Γ(1 − b)Γ(1 + b − c)
4π 2
=
Z (0+,1+,0−,1−)
×
v b−1 (1 − vz)−a (1 − v)c−b−1 dv,
(8.13)
A
where A ∈ (0, 1), z ∈ C \ [1, ∞), b, c − b 6= 1, 2, 3, . . . , and 1/z lies exterior to the contour.
Hence, for α ∈
/ Z and z ∈ C \ [1, ∞),
2 F1 (1
where
− α, α, 1; z) = −
Z (0+,1+,0−,1−)
Y (z; α) =
1
Y (z; α).
4π sin(πα)
(8.14)
v α−1 (1 − vz)α−1 (1 − v)−α dv.
A
We first show that the function Y admits the expansion
Y (θ; α) = (α − n)Y1 + (α − n)2 Y2 + O((α − n)3 ),
α → n,
(8.15)
where Y1 and Y2 are given by (8.11) and (8.12). Let A ∈ (0, 1). Then
Y (z) = (1 − e−2πiα )
Z
L1A
+e2πi(α−1) (1 − e−2πiα )
+ e−2πiα (e2πi(α−1) − 1)
Z
L4A
Z
L2A
+(e2πi(α−1) − 1)
Z
L3A
v α−1 (1 − vz)α−1 (1 − v)−α dv.
Expansion around α = n gives (cf. the proof of (9.11)) equation (8.15) with Y2 given by
(8.12) and
Z
Y1 = 2πi
L1A +L2A +L3A +L4A
y0 dv.
Since y0 is analytic at v = 0 and has a pole at v = 1, we see that Y1 can be expressed as
in (8.11). This proves (8.15).
Equations (8.8) and (8.14) give
"
πiα
#
π2α+1
πα
e− 2
hf (θ) = −
sin
sin2α−2 (θ) Re
Y (z; α) .
ĉ
2
4π sin(πα)
As α → n, we have
ĉ−1 =

−
an
hn
+ α−n
+ O(1),
(α−n)2
h
n
−
α−n + bn + O(α − n),
n = 2, 4, . . . ,
n = 3, 5, . . . ,
where an , bn ∈ R are real constants. We also have
2α+1 = 2n+1 (1 + (α − n) ln 2 + O((α − n)2 ),
51
(8.16)
sin
πα
2
=

n
 (−1) 2 π
2
(−1)
θ = sin
2n−2
+ O((α −
1
n)2 ),
n = 2, 4, . . . ,
n = 3, 5, . . . ,
(θ ) 1 + 2 ln(sin θ )(α − n) + O((α − n)2 ) ,
πiα
πin
πi
e− 2 = e− 2 1 − (α − n) + O((α − n)2 ) ,
2
1
(−1)n
= 2
+ O(α − n),
4π sin(πα)
4π (α − n)
sin
2α−2 1
(α − n) + O((α − n)3 ),
n−1
2
1
Substituting the above expansions into (8.16) and using (8.15), we obtain, if n > 2 is even,
hf (θ) =
2n−2 hn sin2n−2 (θ1 )Re Y1
+ 2n−3 hn sin2n−2 (θ1 )
α−n
an
+ 2 ln(sin θ1 ) Y1 + O(α − n),
× Re 2Y2 − iπY1 + 2 ln 2 −
hn
(8.17)
while, if n > 2 is odd,
2n−1 hn sin2n−2 (θ1 )Im Y1 2n−2 hn sin2n−2 (θ1 )
+
π(α − n)
π
bn
1
+ 2 ln(sin θ ) Y1 + O(α − n).
× Re 2iY2 + πY1 + 2i ln 2 −
hn
hf (θ) = −
(8.18)
In order to establish (8.10), it is therefore enough to show that Re Y1 = 0 for even n and
that Im Y1 = 0 for odd n.
Consider the function J(z) defined by
Z
v n−1 (1 − vz)n−1 (1 − v)−n dv,
J(z) =
|v−1|=
z ∈ C \ {1},
where > 0 is so small that 1/z lies outside the contour. Then, by (9.16),
J(z) = −
Z
v n−1 (1 − vz̄)n−1 (1 − v)−n dv = −J(z̄),
|v−1|=
z ∈ C \ {1}.
Letting u = 1 − v, we can express J(z) as
J(z) = −
Z
(1 − u)n−1 (1 − (1 − u)z)n−1 u−n du.
|u|=
The change of variables u =
J(z) = (−1)n
Z
z−1
z ũ
then yields
(z − z ũ + ũ)n−1 (1 − ũ)n−1 ũ−n dũ = (−1)n−1 J(1 − z),
|ũ|=
z ∈ C \ {0, 1}.
Hence, if Re z = 1/2,
J(z) = −J(z̄) = −J(1 − z) = (−1)n J(z).
Since
Y1 (θ; n) = 2πiJ
1 − i cot θ ,
2
it follows that Re Y1 = 0 (Im Y1 = 0) for even (odd) n. This completes the proof of the
lemma.
52
Taking ξ → 0+ in Theorem 2.6, we obtain the following result for SLEκ (2) in the
fusion limit.
Theorem 8.6 (Green’s function for fused SLEκ (2)). Let 0 < κ 6 4 and consider chordal
SLEκ (2) started from (0, 0+). Then, for each z = x + iy ∈ H,
lim d−2 P2 (Υ∞ (z) 6 ) = c∗ Gf (z),
(8.19)
→0
where P2 is the SLEκ (2) measure, the function Gf is defined by
Gf (z) = (Im z)d−2 hf (arg z),
z ∈ H,
(8.20)
and the constant c∗ = c∗ (κ) is given by (2.11).
For any given integer n > 2, we can compute the integrals in (8.11) and (8.12) defining
Y1 and Y2 explicitly by taking the limit → 0. For the first few simplest cases n = 2, 3, 4,
this leads to the expressions for the fused SLEκ (2) Green’s function presented in the
following proposition.
Proposition 8.7. For α = 2, 3, 4 (corresponding to κ = 4, 8/3, 2, respectively), the function hf (θ) in (8.20) is given explicitly by
hf (θ) =

2


 π (sin θ − θ cos θ) sin θ,
8
2
15π (4θ − 3 sin 2θ + 2θ cos 2θ) sin θ,


 1 (27 sin θ + 11 sin 3θ − 6θ(9 cos θ +
12π
cos 3θ)) sin3 θ,
α = 2,
α = 3,
α = 4,
0 < θ < π.
Proof. The proof relies on long but straightforward computations and is similar to that
of Proposition 9.5.
Remark 8.8. The formulas in Proposition 8.7 can also be obtained by taking the limit
θ2 ↓ θ1 in the formulas of Proposition 9.5.
In view of Lemma 2.9, it follows from Theorem 2.6 that the Green’s function for two
fused commuting SLEs started from 0 is given by the symmetrized expression Gf (z) +
Gf (−z̄). We formulate this as a corollary.
Corollary 8.9 (Green’s function for two fused commuting SLEs). Let 0 < κ 6 4. Consider a system of two fused commuting SLEκ paths in H started from 0 and growing
towards ∞. Then, for each z = x + iy ∈ H,
lim d−2 P (Υ∞ (z) 6 ) = c∗ (Gf (z) + Gf (−z̄)),
→0
where d = 1 + κ/8, the constant c∗ = c∗ (κ) is given by (2.11), and the function Gf is
defined by (8.20).
9
The function G(z, ξ 1 , ξ 2 ) when α is an integer
In Section 2, we defined the function G(z, ξ 1 , ξ 2 ) for noninteger values of α = 8/κ > 1
by equation (2.8). We then claimed that G can be extended to integer values of α by
continuity. The purpose of this section is to verify this claim and to provide formulas for
G(z, ξ 1 , ξ 2 ) in the case when α is an integer.
53
9.1
A representation for h
Equations (2.7) and (2.8) express G in terms of an integral with a Pochhammer contour
enclosing the variable points ξ 2 and z. In order to perform asymptotic calculations (and
in order to make contact with the theory of hypergeometric integrals), it is convenient to
express G in terms of an integral whose contour encloses the fixed points 0 and 1. This
can easily be achieved by means of a linear fractional transformation which maps z and
ξ 2 to 0 and 1, respectively. Moreover, instead of considering G directly, it is convenient to
work with the associated scale invariant function h(θ1 , θ2 ) defined in (6.11).
Lemma 9.1 (Representation for h). Define the function F (w1 , w2 ) by
Z (0+,1+,0−,1−)
F (w1 , w2 ) =
α
α
v α−1 (v − w1 )α−1 (v − w2 )− 2 (1 − v)− 2 dv,
A
w1 , w2 ∈ C \ [0, ∞),
(9.1)
where A ∈ (0, 1) is a basepoint and w1 , w2 are assumed to lie outside the contour. For
each noninteger α > 1, the function h defined in (6.11) admits the representation
h(θ1 , θ2 ; α) =
h
i
sinα−1 θ1
2
Im σ(θ2 )(−eiθ )α−1 F (w1 , w2 ) ,
ĉ
(θ1 , θ2 ) ∈ ∆,
(9.2)
where w1 and w2 are given by
2
2
w1 := 1 − e−2iθ ,
w2 :=
1 − e−2iθ
sin θ2 −i(θ2 −θ1 )
=
e
,
sin θ1
1 − e−2iθ1
(9.3)
the constant ĉ is defined in (2.9), and
(
2
σ(θ ) =
e−iπα ,
eiπα ,
θ2 > π2 ,
θ2 < π2 .
(9.4)
Remark 9.2. For (θ1 , θ2 ) ∈ ∆, w1 and w1 /w2 lie on the circle of radius one centered at
1 (see Figure 8), while w2 lies in the open lower half-plane, i.e., Im w2 < 0.
Remark 9.3. The value of F (w1 , w2 ) in (9.2) is, strictly speaking, not well-defined by
(9.1) for θ2 = π/2, because in this case w2 = 2. However, by analytic continuation, the
function F in (9.2) extends to a multiple-valued function of w1 , w2 ∈ C \ {0, 1}. Equation
(9.2) then extends continuously across the line θ2 = π/2.
Proof. Introducing the new variable v =
1
2
Z (0+,1+,0−,1−)
I(z, ξ , ξ ) =
u−z
ξ 2 −z
in (2.7), we find
(v(ξ 2 − z))α−1 (z − z̄ + v(ξ 2 − z))α−1
A
α
α
× (z + v(ξ 2 − z) − ξ 1 )− 2 ((ξ 2 − z)(1 − v))− 2 (ξ 2 − z)dv
(
2
= (ξ − z)
α−1
F (w1 , w2 ) ×
54
1,
e2iπ(α−1) ,
x 6 ξ2,
x > ξ2,
(9.5)
w1
w2
w1
2θ2
2θ1
0
1
2
1
Figure 8. The complex numbers w1 = 1 − e−2iθ and w1 /w2 = 1 − e−2iθ lie on the circle
of radius one centered at 1.
where
ξ+z
z−ξ
and
z−z̄
z−ξ
are not enclosed by the contour, and the variables
w1 =
z − z̄
2i
=
,
z − ξ2
cot θ2 + i
w2 =
ξ1 − z
cot θ1 + i
=
,
2
ξ −z
cot θ2 + i
can be expressed as in (9.3). The extra factor of e2iπ(α−1) in (9.5) which is present for
x > ξ 2 arises from the factor (z − z̄ + v(ξ 2 − z))α−1 as follows. Let v belong to the
contour in (9.5). Then the complex number z − z̄ + v(ξ 2 − z) lies in the upper half-plane.
If π2 6 θ2 < π (i.e. if x 6 ξ 2 ), then v − w1 also lies in the upper half-plane, but if
0 < θ2 < π/2 (i.e. if x > ξ 2 ), then v − w1 has crossed the negative real axis into the lower
half-plane. The factor e2πi(α−1) is inserted to compensate for this crossing of the branch
cut.
Equations (2.8), (6.11), and (9.5) give
1
h(θ1 , θ2 ) = y α−1 |z − ξ 1 |1−α |z − ξ 2 |1−α |z − ξ 1 |1−α |z − ξ 2 |1−α Im e−iπα I(z, ξ 1 , ξ 2 )
ĉ
1
= sinα−1 (θ1 ) sinα−1 (θ2 )Im σ(θ2 )(− cot θ2 − i)α−1 F (w1 , w2 ) ,
(9.6)
ĉ
where σ is given by (9.4). The representation (9.2) follows.
Let 1{θ2 > π } denote the function which equals 1 if θ2 > π/2 and 0 otherwise. Let
2
LjA := LjA (), j = 1, . . . , 4, be the contours defined in (8.9).
Lemma 9.4. For each (θ1 , θ2 ) ∈ ∆, h(θ1 , θ2 ; α) extends to a continuous function of
55
α ∈ (1, ∞) such that
(
1
2
h(θ , θ ; n) = hn sin
α−1 1
θ ×
2
Im e(n−1)iθ F2 + i(θ2 − 2π1{θ2 > π } )F1 ,
n = 2, 4, . . . ,
2
Im e(n−1)iθ F1 ,
n = 3, 5, . . . ,
(9.7)
2
where the constant hn ∈ R is defined by
hn =
 n n
i Γ( )Γ(n)

− 3 23n
,
2π Γ( 2 −1)
in+1 Γ( n
)Γ(n)

2
− 2 3n
,
4π Γ( 2 −1)
n = 2, 4, . . . ,
(9.8)
n = 3, 5, . . . ,
and the coefficients Fj := Fj (θ1 , θ2 ; n), j = 1, 2, are defined as follows: Let w1 and w2 be
given by (9.3) and define fj := fj (v, θ1 , θ2 ; n), j = 0, 1, by
n
n
n
n
f0 = v n−1 (v − w1 )n−1 (v − w2 )− 2 (1 − v)− 2 ,
f1 = v n−1 (v − w1 )n−1 (v − w2 )− 2 (1 − v)− 2
ln(v − w2 ) ln(1 − v)
−
.
2
2
ln v + ln(v − w1 ) −
Then
 2
n +1
n
n
−1 2
 4π (−1)
2
v n−1 (v − w1 )n−1 (v − w2 )− 2 ,
n
( 2 −1)! ∂v
v=1
F1 =
R

2πi
L30 ()−L40 ()
n = 2, 4, . . . ,
f0 dv,
(9.9)
n = 3, 5, . . . .
and
F2 = −2π 2
Z
Z
L30 ()
f0 dv + 2πi
|v−1|=
f1 dv,
n = 2, 4, . . . ,
(9.10)
where > 0 is so small that w1 , w2 lie exterior to the contours and the principal branch
is used for all complex powers in the integrals.
Proof. Let n > 2 be an integer. We first show that the function F defined in (9.1) admits
the expansion
F (θ1 , θ2 ; α) = (α − n)F1 + (α − n)2 F2 + O((α − n)3 ),
α → n,
(9.11)
where F1 and F2 are given by (9.9) and (9.10). Define f := f (v, w1 , w2 ; α) by
α
α
f = v α−1 (v − w1 )α−1 (v − w2 )− 2 (1 − v)− 2 .
Let > 0 be small and fix A ∈ (, 1 − ). Then we can rewrite (9.1) as
Z
F (w1 , w2 ) =
2πiα
L1A −L3A
= (1 − e
+e
−πiα
Z
L2A +L3A
Z
)
πiα
L1A
+e
2πiα
+e
Z
L2A
Z
L4A −L2A
+e
−πiα
πiα
f (v)dv + (e
56
Z
f (v)dv
−L1A −L4A
−πiα
−e
πiα
) e
Z
L3A
Z
+
L4A
f (v)dv,
where the principal branch is used for all complex powers in all integrals. Since the integral
of f converges at v = 0, we can take → 0 in the first term on the right-hand side, which
gives
F (w1 , w2 ) = (1 − e−πiα )(e2πiα − 1)
Z A
f (v)dv + (eπiα − e−πiα ) eπiα
Z
0
L3A
Z
+
L4A
f (v)dv.
(9.12)
As α → n, we have
(1 − e−πiα )(e2πiα − 1) = 2iπ(−1)n ((−1)n − 1) (α − n) − 2π 2 (α − n)2 + O((α − n)3 ),
eπiα − e−πiα = 2i(−1)n π(α − n) + O((α − n)3 ),
(eπiα − e−πiα )eπiα = 2iπ(α − n) − 2π 2 (α − n)2 + O((α − n)3 ),
and
α
α
f (v) = e(α−1) ln v e(α−1) ln(v−w1 ) e− 2 ln(v−w2 ) e− 2 ln(1−v) = f0 + (α − n)f1 + O((α − n)2 ).
Substituting these expansions into (9.12), we obtain the expansion (9.11) with F1 and F2
given for n > 2 by
n
F1 = 2πi(1 − (−1) )
Z A
Z
f0 dv + 2πi
L3A
0
n
+(−1)
Z
L4A
f0 dv,
and
n
F2 = 2πi(1 − (−1) )
Z A
f1 dv − 2π
0
2
Z A
Z
+
0
L3A
Z
f0 dv + 2πi
L3A
n
Z
+(−1)
L4A
f1 dv.
The expression (9.9) for F1 follows immediately if n is odd. If n is even, then f0 has a
pole of order n/2 at v = 1. Thus, choosing A = 1 − and using the residue theorem, we
find
Z
F1 (w1 , w2 ) = 2πi
|v−1|=
f0 dv
n
v n−1 (v − w1 )n−1 (v − w2 )− 2
= (2πi) Res
,
n
v=1
(1 − v) 2
2
n = 2, 4, . . . ,
(9.13)
which yields the expression (9.9) for F1 also for even n. Finally, letting A = 0, we find
the expression (9.10) for F2 for n even. This completes the proof of (9.11).
We next claim that, as α → n,
2
Im σ(θ2 )(−eiθ )α−1 F (θ1 , θ2 ; α)

−Im e(n−1)iθ2 F2 + i(θ 2 − 2π1{θ2 > π } )F1 (α − n)2 + O((α − n)3 ),
2
=
(n−1)iθ2 
2
−Im e
F1 (α − n) + O((α − n) ),
57
n = 2, 4, . . . ,
n = 3, 5, . . . .
(9.14)
Indeed, the expansion (9.11) yields
2
h
Im σ(θ2 )(−eiθ )α−1 F (θ1 , θ2 ; α) = −Im (1 ∓ iπ(α − n) + · · · )e(n−1)iθ
2
× 1 + ln(−eiθ )(α − n) + · · ·
2
F1 (α − n) + F2 (α − n)2 + · · ·
2
i
2
= − Im e(n−1)iθ F1 (α − n) − Im e(n−1)iθ F2 ∓ iπF1 + i(θ2 − π)F1 (α − n)2
+ O((α − n)3 ).
where the upper (lower) sign applies for θ2 > π/2 (θ2 < π/2) and we have used that
2
ln(−eiθ ) = i(θ2 − π) in the last step. Equation (9.14) therefore follows if we can show
that
h
i
2
Im e(n−1)iθ F1 = 0,
n = 2, 4, 6, . . . .
(9.15)
Let n > 2 be even and define g(w) by
2
n
2
n
g(w) = (1 + e−iθ w)n−1 (1 + eiθ w)n−1 (w + cos θ2 − cot θ1 sin θ2 )− 2 w− 2 .
Then, by (9.13),
h
(n−1)iθ2
Im e
i
h
(n−1)iθ2
h
(n−2)iθ2
F1 = Im e
= Im e
Z
f0 dv
2πi
|v−1|=
Z
i
2
2
2
(1 + e−iθ w)n−1 (e−iθ w + e−2iθ )n−1
2πi
|w|=
−iθ2
× (e
n
2
n
(w + cos θ2 − cot θ1 sin θ2 ))− 2 (−e−iθ w)− 2 dw
h
= (−1)−n/2 Im 2πi
Z
g(w)dw
i
i
|w−1|=
2
where we have used the change of variables v = 1 + e−iθ w and the definitions (9.3) of w1
and w2 in second equality. Since g(w) = g(w̄), the identity
Z
Z
g(w)dw =
γ
g(v̄)dv
(9.16)
γ̄
R
valid for a general contour γ ⊂ C, implies that |w−1|= g(w)dw is pure imaginary. This
proves (9.15) and hence also (9.14).
For each integer n > 2, we have the following asymptotic behavior of ĉ−1 as α → n:

−
hn
1
+ O α−n
, n even,
1
(α−n)2
=
ĉ − hn + O(1),
n odd,
α−n
n = 2, 3, 4, . . . .
(9.17)
Substituting (9.14) and (9.17) into (9.2), we find (9.7).
By taking the limit as approaches zero in the integrals in (9.9) and (9.10), it is
possible to derive explicit expressions for F1 and F2 , and hence also for the function
h. We illustrate this by computing the SLEκ (2) Green’s function explicitly for κ = 4,
κ = 8/3, and κ = 2.
58
Proposition 9.5. For κ = 4, κ = 8/3, and κ = 2 (i.e. for α = 2, 3, 4), the SLEκ (2)
Green’s function is given by equation (2.12) where h(θ1 , θ2 ) is given explicitly by
h(θ1 , θ2 ) =
h(θ1 , θ2 ) =
1
sin(2θ1 − 2θ2 ) + 2θ1 (1 − cos 2θ2 ) + 2θ2 (cos 2θ1 − 1)
4π sin(θ1 − θ2 )
− sin 2θ1 + sin 2θ2 ,
κ = 4,
(9.18)
√
θ1 − 3θ2
1
sin θ1 sin θ2 − 6 cos
1
2
30π(cos(θ − θ ) + 1)
2
1
1
1
1
2
2
5θ − 3θ
3θ − θ2
θ + θ2
3θ − 5θ
+ cos
− 6 cos
− 38 cos
+ cos
2
2
2
2
1
1
1
2
2
2
3θ + 3θ
5θ + θ
θ + 5θ
+ 20 cos
+ 14 cos
+ 14 cos
2
2
2
θ1 − θ2
2
− 2 cos2
!
− 9 sin 2θ1 sin 2θ2 + (7 cos 2θ2 + 8) cos 2θ1 + 8 cos 2θ2
θ1 + θ2
− 23 arg cos
2
!
!
√
1
2
,
+ i sin θ sin θ
κ = 8/3,
(9.19)
and
h(θ1 , θ2 ) =
1
72 sin5 (θ1 ) cos(θ2 ) cos(θ1 − 3θ2 )
sin3 (θ2 )
+
192π
(cot θ1 − cot θ2 )3
sin3 (θ1 − θ2 )
(3θ1 cot θ2 + 2) cot θ1 + θ1 (3 − 2 csc2 θ1 ) − 3 cot θ2
× 96
sin θ1
+ csc6 (θ2 ) 3 16θ2 (3 sin(θ1 − 2θ2 ) + sin θ1 ) sin θ1 + 5 sin 2θ2 − 4 sin 4θ2 sin θ1
+ 6 cos(θ1 − 6θ2 ) − cos(3θ1 − 6θ2 ) + (75 cos 2θ2 − 30 cos 4θ2 − 33) cos θ1
− 17 cos 3θ1
,
κ = 2.
(9.20)
Proof. We give the proof for κ = 4. The proofs for κ = 8/3 and κ = 2 are similar. Let
n = 2. As goes to zero, we have
Z
Z
L30 ()
f0 dv =
L30 ()
v(v − w1 )
dv
(v − w2 )(1 − v)
w2 (w1 − w2 ) ln(v − w2 ) − (w1 − 1) ln(1 − v) + v(1 − w2 ) 1+−i0
w2 − 1
v=0
−(w1 − 1) ln + J1 (w1 , w2 ) + O(),
=
w2 − 1
=
where the order one term J(w1 , w2 ) is given by
J1 (w1 , w2 ) =
w2 (w1 − w2 )(ln(1 − w2 ) − ln(−w2 )) − iπ(w1 − 1) − w2 + 1
.
w2 − 1
59
On the other hand, since the function
ln v + ln(v − w1 ) −
ln(v − w2 )
2
is analytic at v = 1, the residue theorem gives
v(v − w1 )
ln(v − w2 ) ln(1 − v)
f1 dv =
ln v + ln(v − w1 ) −
dv
−
(v
−
w
)(1
−
v)
2
2
|v−1|=
|v−1|=
2
1 − w1
ln(1 − w2 )
= − 2πi
ln 1 + ln(1 − w1 ) −
1 − w2
2
Z 2π
iϕ
iϕ
1
(1 + e )(1 + e − w1 )
−
ln(−eiϕ )ieiϕ dϕ
2 0
(1 + eiϕ − w2 )(−eiϕ )
ln(1 − w2 )
1 − w1
ln(1 − w1 ) −
= − 2πi
1 − w2
2
Z 2π
i
1 − w1
+
(ln + i(ϕ − π))dϕ + O( ln )
2 0 1 − w2
1 − w1
ln + J2 (w1 , w2 ) + O( ln ),
= iπ
1 − w2
Z
Z
where the order one term J2 (w1 , w2 ) is given by
1 − w1
ln(1 − w2 )
J2 (w1 , w2 ) = −2πi
ln(1 − w1 ) −
.
1 − w2
2
Hence, since the singular terms of O(ln ) cancel,
F2 = 2πi lim πi
→0
Z
L30
Z
f0 (v)dv +
|v−1|=
= 2πi πiJ1 (w1 , w2 ) + J2 (w1 , w2 ) ,
f1 (v)dv
n = 2.
(9.21)
On the other hand,
2
v(v − w1 )
1 − w1
4π 2 e−iθ sin θ1
= −(2πi)2
=
,
v=1 (v − w2 )(1 − v)
1 − w2
sin(θ1 − θ2 )
F1 = (2πi)2 Res
n = 2. (9.22)
The terms J1 and J2 involve the logarithms ln(1 − w1 ), ln(1 − w2 ), and ln(−w2 ). The
expressions (9.3) for w1 and w2 imply that (recall that principal branches are used for all
logarithms)
ln(1 − w1 ) = ln e−2iθ
2
= 2i π1{θ2 > π } − θ2 ,
2
sin(θ 2 − θ 1 ) sin(θ2 − θ1 )
+ i(π − θ 2 ),
ln(1 − w2 ) = ln − e
=
ln
sin θ1
sin θ1
2
sin θ 2 2
1 sin θ
1
2
ln(−w2 ) = ln − e−i(θ −θ )
=
ln
sin θ 1 + i(π + θ − θ ),
sin θ1
−iθ2
60
(9.23)
for all (θ1 , θ2 ) ∈ ∆. For n = 2, equation (9.7) gives
h(θ1 , θ2 ; 2) =
h 2
i
sin θ1
iθ
2
Im
e
F
+
i(θ
−
2π1
.
2 > π } )F1
2
{θ
2
2π 3
Substituting the expressions (9.21) and (9.22) into this formula and using (9.23), equation
(9.18) follows after simplification.
10
Asymptotics of Coulomb gas integrals
The goal of this section is to suggest a methodology for rigorously establishing (at least
some) important observables for SLE with multiple curves and/or insertions. As explained
in Section 4, candidates for these observables can be obtained by applying a screening procedure to the local martingales of the form (4.2) obtained from conformal field theory.
A key step in the analysis consists of choosing the appropriate screening integration contour. This contour is determined by the requirement that the associated local martingale
satisfies the correct boundary conditions. For the Schramm observable for SLEκ (2), it
will be directly verified in Appendix A that the function P (z, ξ) defined in (2.5) fulfills
the correct boundary conditions. (This can also be proved using the method developed
in this section.) For the Green’s function, the analogous verification is more complicated
and involves the asymptotic analysis of the integral (2.7) as two or more of the points
z, z̄, ξ 1 , ξ 2 come together. In this section, we present an approach for computing asymptotics of this type to all orders. Subsequently, in Appendix B, we use this approach to
verify that the function (2.8) satisfies the boundary conditions expected from the SLEκ (2)
Green’s function.
10.1
Screening integrals
In the case of one screening integral, the screening procedure (see Section 4) gives rise to
local SLE martingales involving expressions of the form
Z (z1 +,z2 +,z1 −,z2 −) Y
N
A
(u − zj )aj du,
(10.1)
j=1
N
where {zj }N
1 ⊂ C is a finite collection of points, {aj }1 ⊂ R is a set of exponents, and
the Pochhammer integration contour encloses two of these points, say z1 and z2 . In order
to find the boundary behavior of the local martingale, it is necessary to compute the
asymptotics of the integral (10.1) as two or more of the zj merge. This presents some
difficulties, because if a point zk , k = 3, . . . , N , approaches z1 or z2 , then the contour gets
squeezed between the two points which, in general, gives rise to singular behavior.
By applying a linear fractional transformation, we may assume that z1 = 0 and z2 = 1
in (10.1); this yields an expression of the form
Z (0+,1+,0−,1−)
v a1 (1 − v)a2
A
N
Y
(v − wj )aj dv.
j=3
61
(10.2)
For N = 3, this expression defines a hypergeometric function and the asymptotics can
be obtained from well-known identities and expansions of these functions. However, for
N > 4, the analysis is more intricate and we have not been able to find a reference dealing
with this case. In this section, we consider the class of integrals corresponding to N = 4.
More precisely, we consider asymptotics of the function F (w1 , w2 ) := F (a, b, c, d; w1 , w2 )
defined by
Z (0+,1+,0−,1−)
F (w1 , w2 ) =
v a (v − w1 )b (v − w2 )c (1 − v)d dv,
A
(w1 , w2 ) ∈ D0 ⊂ C2 ,
(10.3)
where A ∈ (0, 1) is a base point, a, b, c, d ∈ R are real exponents, and w1 , w2 are assumed
to lie outside the contour. In order to make F single-valued, we have restricted the domain
of definition in (10.3) to the domain D0 ⊂ C2 defined by
D0 = {(w1 , w2 ) ∈ C2 | w2 ∈ C \ [0, ∞), w1 ∈ C \ ([0, ∞) ∪ γ(w2 ,∞) )},
where γ(w2 ,∞) ⊂ C denotes a branch cut from w2 to ∞ (to be specific, we henceforth
choose γ(w,∞) = {rw | r > 1}). The function F defined in (9.1) is the special case of (10.3)
when
α
c=d=− .
2
a = b = α − 1,
(10.4)
It will be clear from the analysis that a similar approach can be used also for N > 5.
We assume that a, d ∈ R \ Z are not integers, because otherwise F vanishes identically.
We note that F can be analytically continued to a multiple-valued analytic function
of w1 ∈ C \ {0, 1, w2 } and w2 ∈ C \ {0, 1}. We will compute the asymptotic behavior of
F to all orders as one or both of the points w1 and w2 approach 0 or 1. The basic idea is
the following: If we want to consider the limit w1 → 0 say, then we rewrite F as a sum
of two terms. One term which is defined by the same integral as F except that w1 is now
assumed to lie inside the contour in the same component as 0, and a second term which
is defined by a similar expression but with the Pochhammer contour enclosing {0, w1 }
instead of {0, 1}, see equation (10.24). The asymptotics of both of these terms can easily
be computed to all orders by replacing (v − w1 )b by its asymptotic expansion as w1 → 0:
b
(v − w1 ) ∼
∞
X
(−1)k
k=0
Γ(b + 1)
wk
v b−k 1 .
Γ(b + 1 − k)
k!
(10.5)
We emphasize that we cannot, in general, compute the asymptotics of F as w1 → 0
by substituting the expansion (10.5) directly into (10.3). Indeed, such a procedure gives
the correct contribution from the first term, but completely ignores the contribution from
the second term.
10.2
The hypergeometric case of N = 3
Before turning to the case N = 4, let us consider, as motivation, the case N = 3 in which
the integral in (10.2) reduces to a hypergeometric function.
62
Let F (w) denote the expression in (10.2) for N = 3:
Z (0+,1+,0−,1−)
F (w) =
v a (v − w)c (1 − v)d dv,
A
w ∈ C \ [0, ∞).
(10.6)
If F̃ denotes the function
Z (0+,1+,0−,1−)
F̃ (w) =
v a (w − v)c (1 − v)d dv,
A
w ∈ C \ (−∞, 1],
where w lies exterior to the contour, then the definition (8.13) of 2 F1 implies
F̃ (w) =
4π 2 wc 2 F1 (−c, a + 1, a + d + 2; 1/w)
,
e−(a+d+2)πi Γ(a + d + 2)Γ(−a)Γ(−d)
w ∈ C \ (−∞, 1].
(10.7)
On the other hand, F and F̃ are related by
F (w) = ρ(w)F̃ (w),
w1 , w2 ∈ C \ R,
where
(
ρ(w) =
e−iπc ,
eiπc ,
Im w > 0,
Im w < 0.
(10.8)
Thus we can use the relation (10.7) to derive asymptotic expansions of F (w) as w → 0
and w → 1. For definiteness, let us consider the limit w → 1. The function 2 F1 is an
analytic function of z with a branch cut along [1, ∞); in particular, it is not analytic at
z = 1. In order to arrive at the correct asymptotics as w → 1, we therefore first use the
hypergeometric identity (Eq. (10.12) in [34])
2 F1 (a, b, c; z)
=
Γ(c)Γ(c − a − b)
2 F1 (a, b, 1 + a + b − c; 1 − z)
Γ(c − a)Γ(c − b)
Γ(c)Γ(a + b − c)
+
(1 − z)c−a−b 2 F1 (c − a, c − b, 1 + c − a − b; 1 − z),
Γ(a)Γ(b)
z ∈ C \ ((−∞, 0] ∪ [1, ∞)), (10.9)
to rewrite equation (10.7) as
4πeiπ(a+d+2) sin(πa)Γ(a + 1)wc sin(πd) csc(π(c + d))
Γ(−c − d)Γ(a + c + d + 2)
1
× 2 F1 1 + a, −c, −c − d; 1 −
w
F̃ (w) = −
4πeiπ(a+d+2) sin(πa) sin(πd)Γ(d + 1) c w − 1 c+d+1
w
Γ(−c)Γ(c + d + 2) sin(π(c + d))
w
1
,
w ∈ C \ (−∞, 1].
× 2 F1 d + 1, a + c + d + 2; c + d + 2; 1 −
w
+
63
(10.10)
The hypergeometric functions in (10.10) are analytic at w = 1. In fact,
2 F1 (a, b, c; z)
=
∞
X
(a)k (b)k z n
k=0
(c)k
n!
,
where the Pochhammer symbol (a)k is defined by
(a)k =
Γ(a + k)
= (a + k − 1) · · · (a + 2)(a + 1)a.
Γ(a)
Using this expansion in (10.10), we find the following representation which gives the
asymptotics of F̃ (w) as w → 1 to all orders:
F̃ (w) = P1 (w) + (w − 1)c+d+1 P2 (w)
(10.11)
P1 (w) = (−1 + e2πia − e2πi(a+d) + e2πid )P̂1 (w),
(10.12a)
P2 (w) = (−1 + e2πia − e2πi(a+d) + e2πid )P̂2 (w),
(10.12b)
where
and
P̂1 (w) =
P̂2 (w) =
∞
X
πΓ(a + 1)Γ(c + 1) csc((c + d + 1 − k)π) (w − 1)k
,
Γ(c + 1 − k)Γ(k − c − d)Γ(2 + a + c + d − k)
k!
k=0
∞
X
πΓ(a + 1)Γ(d + 1 + k) csc((c + d)π) (w − 1)k
k=0
10.3
Γ(a + 1 − k)Γ(−c)Γ(c + d + 2 + k)
k!
.
Asymptotics of F as w2 → 1
Let us now consider the case of N = 4 in which F (w1 , w2 ) is given by (10.3). We first
consider the asymptotics of F as w2 → 1. The basic idea is to derive generalizations of
the hypergeometric identities (10.9) and (10.11).
Let D1 ⊂ C2 denote the domain
D1 = {(w1 , w2 ) ∈ C2 | w1 ∈ C \ [0, ∞), w2 ∈ C \ ((−∞, 1] ∪ γ(w1 ,∞) )},
(10.13)
and define F̃ by
Z (0+,1+,0−,1−)
F̃ (w1 , w2 ) =
v a (v − w1 )b (w2 − v)c (1 − v)d dv,
(w1 , w2 ) ∈ D1 , (10.14)
A
where w1 , w2 lie exterior to the contour. Then
F (w1 , w2 ) = ρ(w2 )F̃ (w1 , w2 ),
where ρ is the function in (10.8).
64
(w1 , w2 ) ∈ D0 ∩ D1 ,
(10.15)
w1
w2
A
0
Re v
1
Figure 9. In the definition of the function P1 , the point w1 lies exterior to the contour,
whereas w2 lies inside the contour in the same component as 1.
Assuming that c + d ∈
/ Z, we define two functions Pj : D1 → C, j = 1, 2, as follows.
The function P1 is defined (up to a constant) by the same formula as F̃ except that the
point w2 is assumed to lie inside the contour in the same component as 1; more precisely,
P1 (w1 , w2 ) =
e−iπc sin(dπ)
sin(π(d + c))
Z (0+,1+,0−,1−)
v a (v − w1 )b (w2 − v)c (1 − v)d dv,
(w1 , w2 ) ∈ D1 ,
A
where w1 lies outside the contour and w2 lies inside the contour in the same component as
1, see Figure 9. The function P2 : D1 → C is defined as follows. First, given w1 ∈ C\[0, ∞),
we define P2 (w1 , w2 ) for Re w2 ∈ (0, 1) with Im w2 > 0 sufficiently small by
P2 (w1 , w2 ) =
×
eiπ(a+d) sin(aπ) −iπ(c+d+1)
e
sin(π(d + c))
Z (0+,1+,0−,1−)
(w2 + s(1 − w2 ))a (w2 − w1 + s(1 − w2 ))b sc (1 − s)d ds,
(10.16)
A
2
1 −w2
where A ∈ (0, 1) and the points ww2 −1
and w1−w
are assumed to lie exterior to the contour.
2
Then, for each w1 ∈ C \ [0, ∞), we use analytic continuation to extend P2 to a (singlevalued) analytic function of w2 ∈ C \ ((−∞, 1] ∪ γ(w1 ,∞) ). The latter step is permissible
2
because the function P2 can be analytically continued as long as the points ww2 −1
and
w1 −w2
/ {0, 1, w1 , ∞}.
1−w2 stay away from the set {0, 1, ∞}, i.e., as long as w2 ∈
Let f (w1 , w2 ±i0) denote the boundary values of a function f (w1 , w2 ) as w2 approaches
the real axis from above and below, respectively.
Lemma 10.1. Suppose a, d, c + d ∈
/ Z. Then
F̃ (w1 , w2 ) = P1 (w1 , w2 ) + (w2 − 1)c+d+1 P2 (w1 , w2 ),
(w1 , w2 ) ∈ D1 .
(10.17)
Proof. It is enough to show that
F̃ (w1 , w2 + i0) = P1 (w1 , w2 + i0) + |1 − w2 |c+d+1 eiπ(c+d+1) P2 (w1 , w2 + i0),
w1 ∈ C \ [0, ∞), w2 ∈ (0, 1).
(10.18)
Indeed, for each w1 ∈ C\[0, ∞), both sides of the equation (10.17) are analytic functions of
w2 ∈ C \ ((−∞, 1] ∪ γ(w1 ,∞) )} which can be extended to multiple-valued analytic functions
of w2 ∈ C \ {0, 1, w1 }. Hence (10.17) follows from (10.18) by analytic continuation.
65
Let us prove (10.18). Let > 0 be small. Let w1 ∈ C \ [0, ∞) and w2 ∈ (0, 1). Given
w ∈ C, let
Sw− = {w + eiφ | − π 6 φ 6 0},
Sw+ = {w + eiφ | 0 6 φ 6 π},
denote counterclockwise semicircles of radius centered at w. Here and below we adopt
the convention that unless the integration contour is a Pochhammer contour, the principal
branch is used for all complex powers and logarithms in the integrand. Then we can write
Z
F̃ (w1 , w2 + i0) =
Z
+e2πia
L1w
L2w
2 −
2 −
−
+Sw
+L3w
2
Z
2πi(a+d)
−e
2πid
Z2
+e
−
+L2w
Sw
2
−
L3w
2
−
+ +Sw2
Z
L4w
+
2 +
2πi(d+c)
−L1w
2 −
Z
+e2πi(a+d+c)
2 −
−
+Sw
2
−e
Z
L4w
2 +
v a (v − w1 )b (w2 − v)c (1 − v)d dv
and
e−iπc sin(dπ)
P1 (w1 , w2 + i0) =
sin((c + d)π)
− e2πi(d+c)
Z
Z
+e2πia
L1w
L2w
2 −
Z
+
L1w − +Sw
+L4w +
2
2
2
−
2 −
−
+Sw
+L3w
2
+e2πi(a+d+c)
−
L3w + +Sw
2
2
L4w
2 +
Z
Z
2 +
+
+Sw
−L2w
2
v a (v − w1 )b (w2 − v)c (1 − v)d dv.
Simplification gives
F̃ (w1 , w2 + i0) = (1 − e
+ (e
2πid
2πia
Z
)
− 1)
L1w
+e
2πia
L2w
2 −
2 −
Z
L3w
2 +
Z
−
+Sw
2
+e
2πi(d+c)
Z
−e
L4w
2πid
Z
−
Sw
2
2 +
× v a (v − w1 )b (w2 − v)c (1 − v)d dv
and
e2iπd − 1
(1 − e2πi(d+c) )
e2πi(d+c) − 1
P1 (w1 , w2 + i0) =
+ (e
2πia
Z
− 1)
Z
+e2πia
L1w
2 −
2 +
−
+Sw
2
+e
Z
L2w
2 −
Z
2πi(d+c)
L3w
L4w
2 +
+
+Sw
2
× v a (v − w1 )b (w2 − v)c (1 − v)d dv.
Hence
F̃ (w1 , w2 + i0) − P1 (w1 , w2 + i0) =
+
e2πic (e2πid
1−
− 1)
e2πic
(e2πia − 1) sin(cπ)eiπd
sin(π(d + c))
Z
+
−
Sw
+Sw
2
2
66
2 −
Z
L3w
+e2πi(d+c)
2 +
v a (v − w1 )b (w2 − v)c (1 − v)d dv.
Z
L4w
2 +
Using the identity
(
c
(w2 − v) =
e−iπc (v − w2 )c ,
eiπc (v − w2 )c ,
v ∈ Sw+2 ∪ L4w2 + ,
v ∈ Sw−2 ∪ L3w2 + ,
we can write this as
(e2πia − 1) sin(cπ)eiπ(d+c)
F̃ (w1 , w2 + i0) − P1 (w1 , w2 + i0) =
sin(π(d + c))
e2πid − 1
+
1 − e2πic
Z
+
Sw
2
e2πic (e2πid − 1)
+
1 − e2πic
1
,
e2πic −1
2πid
L3w
+e
L4w
2 +
2 +
−
Sw
2
w1 ∈ C \ [0, ∞),
w2 ∈ (0, 1),
we obtain
(e2πia − 1) sin(cπ)eiπ(d+c)
F̃ (w1 , w2 + i0) − P1 (w1 , w2 + i0) =
(e2πic − 1)
sin(π(d + c))(e2πic − 1)
+ e2πid (e2πic − 1)
Z
Z
× v a (v − w1 )b (v − w2 )c (1 − v)d dv,
Factoring out
Z
Z
L4w
+(1 − e2πid )
2 +
Z
+
Sw
2
+e2πic (1 − e2πid )
Z
L3w
2 +
Z
−
Sw
2
× v a (v − w1 )b (v − w2 )c (1 − v)d dv.
That is,
(e2πia − 1) sin(cπ)eiπ(d+c)
F̃ (w1 , w2 + i0) − P1 (w1 , w2 + i0) =
sin(π(d + c))(e2πic − 1)
a
b
Z (w2 +,1+,w2 −,1−)
c
w2 +
d
× v (v − w1 ) (v − w2 ) (1 − v) dv.
Performing the change of variables s =
interval (0, 1), this yields
F̃ (w1 , w2 + i0) − P1 (w1 , w2 + i0) =
v−w2
1−w2 ,
which maps the interval (w2 , 1) to the
ei(a+d)π sin(aπ)
sin(π(d + c))
Z (0+,1+,0−,1−)
(w2 + s(1 − w2 ))a
A
× (w2 + s(1 − w2 ) − w1 )b (s(1 − w2 ))c ((1 − w2 )(1 − s))d (1 − w2 )ds.
Comparing this expression with the definition (10.16) of P2 , equation (10.18) follows.
Using the identity (10.17), we can easily find the asymptotics of F = ρ(w2 )F̃ as w2 → 1
to all orders. Indeed, the functions P1 and P2 in (10.17) admit asymptotic expansions to
all orders as follows. Substituting the expansion
(w2 − v)c ∼ (1 − v)c + c(w2 − 1)(1 − v)c−1 +
=
∞
X
c(c − 1)(w2 − 1)2
(1 − v)c−2 + · · ·
2
Γ(c + 1) (w2 − 1)k
(1 − v)c−k ,
Γ(c
+
1
−
k)
k!
k=0
67
w2 → 1,
into the definition of P1 (w1 , w2 ) and recalling that w2 and 1 lie in the same component
inside the contour, we find
P1 (w1 , w2 ) ∼
∞
e−iπc sin(dπ) X
Γ(c + 1) (w2 − 1)k
sin(π(d + c)) k=0 Γ(c + 1 − k)
k!
×
Z (0+,1+,0−,1−)
v a (v − w1 )b (1 − v)d+c−k dv,
w2 → 1,
(10.19)
A
where the integral on the right-hand side can be expressed in terms of hypergeometric
functions if desired. Similarly, substituting the expansions
(w2 + s(1 − w2 ))a ∼
∞
X
Γ(a + 1)
(w2 − 1)k
(1 − s)k
,
Γ(a + 1 − k)
k!
k=0
w2 → 1,
and
(w2 + s(1 − w2 ) − w1 )b ∼
∞
X
l=0
(w2 − 1)l
Γ(a + 1)
(1 − s)l (1 − w1 )b−l
,
Γ(a + 1 − l)
l!
w2 → 1,
into the definition of P2 (w1 , w2 ), we find, as w2 → 1,
∞ X
∞
eiπ(a+d) sin(aπ) −iπ(c+d+1) X
Γ(a + 1)
Γ(a + 1)
P2 (w1 , w2 ) ∼
e
sin(π(d + c))
Γ(a
+
1
−
k)
Γ(a
+ 1 − l)
k=0 l=0
×
10.4
(w2 − 1)l
(w2 − 1)k
(1 − w1 )b−l
k!
l!
Z (0+,1+,0−,1−)
sc (1 − s)d+k+l ds. (10.20)
A
Asymptotics of F as w1 → 0
In order to determine the asymptotics of F (w1 , w2 ) as w1 → 0, we define two functions
Qj : D0 → C, j = 1, 2, as follows. The function Q1 (w1 , w2 ) is defined by
e2πia − 1
Q1 (w1 , w2 ) = 2πi(a+b)
e
−1
Z (0+,1+,0−,1−)
v a (v − w1 )b (v − w2 )c (1 − v)d dv,
(w1 , w2 ) ∈ D0 ,
A
where A ∈ (0, 1), w1 lies inside the contour in the same component as 0, and w2 lies
outside the contour. Given w2 ∈ C \ [0, ∞), we define Q2 (w1 , w2 ) for Re w1 ∈ (0, 1) with
Im w1 < 0 sufficiently small by
Q2 (w1 , w2 ) =
(e2πid − 1)e−iπb
1 − e−2iπ(a+b)
Z (0+,1+,0−,1−)
sa (1 − s)b (sw1 − w2 )c (1 − sw1 )d ds, (10.21)
A
1
2
where A ∈ (0, 1) and the points w
w1 and w1 lie exterior to the contour. For each w2 ∈
C \ [0, ∞), we then use analytic continuation to extend Q2 to a function of w1 ∈ C \
([0, ∞) ∪ γ(w2 ,∞) ).
Lemma 10.2. Suppose a, d, a + b ∈
/ Z. Then
F (w1 , w2 ) = Q1 (w1 , w2 ) + w1a+b+1 Q2 (w1 , w2 ),
68
(w1 , w2 ) ∈ D0 .
(10.22)
Proof. By analyticity, is enough to show that
F (w1 − i0, w2 ) = Q1 (w1 − i0, w2 ) + w1a+b+1 Q2 (w1 − i0, w2 )
(10.23)
for w1 ∈ (0, 1) and w2 ∈ C \ [0, ∞).
Let > 0 be small. Let w1 ∈ (0, 1) and w2 ∈ C \ [0, ∞). Then
F (w1 − i0, w2 ) =
Z
+e2πi(a+b)
L1w
Z
L2w
1 −
+
Sw
1
Z
+
1 +
−e2πi(a+b+d)
Z
+e2πia
−e2πid
L2w
+
Sw
1
Z
L3w
+e2πi(a+d)
1 +
Z
L4w
1 +
Z
L1w
1 −
Z
L3w
+
Sw
1
1 −
Z
+ e2πi(a+d)
−
Z
−e2πia
1 −
+
+Sw
+L4w
1
1 +
v a (v − w1 )b (v − w2 )c (1 − v)d dv,
where w2 lies exterior to the contours. Moreover,
Q1 (w1 − i0, w2 ) =
e2πia − 1
e2πi(a+b) − 1
Z
+ e2πi(a+b+d)
Z
+e2πi(a+b)
+
Sw
+L1w
Z
L2w
1 −
1 −
−
L4w+ −Sw
−L2w−
−e2πid
−
+Sw
+L3w+
Z
Z
+
+L4w+
L1w− +Sw
+
−L3w+
× v a (v − w1 )b (v − w2 )c (1 − v)d dv.
Simplification gives
F (w1 − i0, w2 ) = (1 − e
2πid
Z
)
2πi(a+b)
(1 − e
Z
Z
+e
L1w
2πid
Z
)
L2w
1 −
+ (e
2πia
a
1 −
2πid
− 1) (e
− 1)
b
+
Sw
1
c
+
2πid
+e
L3w
Z
L4w
1 +
1 +
d
× v (v − w1 ) (v − w2 ) (1 − v) dv
and
e2πia − 1
Q1 (w1 − i0, w2 ) = 2πi(a+b)
(1 − e2πid )
e
−1
+ e2πi(a+b) (1 − e2πid )
+e
2πid
2πi(a+b)
(e
− 1)
Z
−
Sw
Z
+e2πi(a+b) (1 − e2πid )
L1w
1 −
L4w+
+(1 − e
2πid
Z
L3w+
Z
)
+
Sw
× v a (v − w1 )b (v − w2 )c (1 − v)d dv.
Hence
(e2πid − 1)e−2iπb
F (w1 − i0, w2 ) − Q1 (w1 − i0, w2 ) =
(1 − e2iπb )
1 − e−2iπ(a+b)
69
L2w
1 −
+(e2πi(a+b) − 1)
Z
Z
Z
L1w
1 −
+ e2πi(a+b) (1 − e2iπb )
Z
L2w
+e2πib (e2πia − 1)
1 −
Z
+
−
Sw
+Sw
1
1
v a (v − w1 )b (v − w2 )c (1 − v)d dv.
Using the identity
(
b
(v − w1 ) =
eiπb (w1 − v)b ,
v ∈ Sw+ ∪ L1w+ ,
e−iπb (w1 − b)b , v ∈ Sw− ∪ L2w+ ,
we can write this as
(e2πid − 1)e−iπb
F (w1 − i0, w2 ) − Q1 (w1 − i0, w2 ) =
(1 − e2iπb )
1 − e−2iπ(a+b)
+ e2πia (1 − e2iπb )
Z
L2w
+e2πib (e2πia − 1)
1 −
× v a (w1 − v)b (v − w2 )c (1 − v)d dv,
Z
+
Sw
1
+(e2πia − 1)
w1 ∈ (0, 1),
Z
Z
L1w
1 −
−
Sw
1
w2 ∈ C \ [0, ∞).
That is,
(e2πid − 1)e−iπb (0+,w1 +,0−,w1 −)
F (w1 − i0, w2 ) − Q1 (w1 − i0, w2 ) =
1 − e−2iπ(a+b) w1 −
× v a (w1 − v)b (v − w2 )c (1 − v)d dv.
Z
Performing the change of variables s =
(0, 1), we obtain
v
w1 ,
(10.24)
which maps the interval (0, w1 ) to the interval
(e2πid − 1)e−iπb a+b+1 (0+,1+,0−,1−)
w
F (w1 − i0, w2 ) − Q1 (w1 − i0, w2 ) =
1 − e−2iπ(a+b) 1
A
a
b
c
× s (1 − s) (sw1 − w2 ) (1 − sw1 )d ds.
Z
The lemma follows.
In the same way that (10.17) gives the asymptotics as w2 → 1, the identity (10.22)
gives an asymptotic expansion of F (w1 , w2 ) as w1 → 0 to all orders.
10.5
Asymptotics of F as w1 → 0 and w2 → 0
We next determine the asymptotics of F (w1 , w2 ) in the regime where both w1 and w2
approach zero. Assuming a + b, a + b + c ∈
/ Z, we define two functions Rj : D0 → C,
j = 1, 2, as follows. The function R1 (w1 , w2 ) is defined for (w1 , w2 ) ∈ D0 by
e2πia − 1
R1 (w1 , w2 ) = 2πi(a+b+c)
e
−1
Z (0+,1+,0−,1−)
v a (v − w1 )b (v − w2 )c (1 − v)d dv,
A
where A ∈ (0, 1) and both points w1 and w2 are assumed to lie inside the contour in the
same component as 0. For 0 < Re w1 < Re w2 < 1 with Im w1 < 0 and Im w2 < 0, we
define R2 (w1 , w2 ) by
R2 (w1 , w2 ) =
e2πi(a+b) (e2πia − 1)(e2πid − 1)eiπc
(e2πi(a+b) − 1)(e2πi(a+b+c) − 1)
70
×
Z (0+,1+,0−,1−)
sa (sw2 − w1 )b (1 − s)c (1 − sw2 )d ds,
(10.25)
A
w1
where we assume A ∈ (0, 1) is so large that Re (Aw2 − w1 ) > 0, that the point w
lies
2
1
inside the contour in the same component as 0, and that w2 lies outside the contour. We
then use analytic continuation to extend R2 to all of D0 .
Lemma 10.3. Suppose a, d, a + b, a + b + c ∈
/ Z. Then
F (w1 , w2 ) = R1 (w1 , w2 ) + w2a+c+1 R2 (w1 , w2 ) + w1a+b+1 Q2 (w1 , w2 ),
(w1 , w2 ) ∈ D0 .
(10.26)
Proof. Both sides of (10.26) are analytic functions of (w1 , w2 ) ∈ D0 which extend to
multiple-valued analytic functions of
(w1 , w2 ) ∈ C2 \ {w1 = 0} ∪ {w1 = 1} ∪ {w2 = 0} ∪ {w2 = 1} ∪ {w1 = w2 } .
(10.27)
Hence, by Lemma 10.2, it is enough to show that
Q1− (w1 , w2 ) = R1− (w1 , w2 ) + w2a+c+1 R2− (w1 , w2 ),
0 < w1 < w2 < 1,
(10.28)
where, for a function f , we use the short-hand notation f− (w1 , w2 ) := f (w1 − i0, w2 − i0).
Let 0 < w1 < w2 < 1 and suppose 0 < < 21 min{w1 , w2 − w1 , 1 − w2 }. Then
e2πia − 1
Q1− (w1 , w2 ) = 2πi(a+b)
e
−1
+ e2πi(a+b)
Z
Z
L3w
−
L3w +
2
+
+
Sw
2
2πi(a+b)
−e
L2w
Z
L4w
2 +
+
+Sw
2
−e2πi(a+b+c+d)
Z
+
Sw
2
2 −
2 −
+e2πi(a+b+d)
Z
Z
+e
L1w
2 +
Z
2πi(a+b+c)
Z
−e2πid
L2w
Z
2 −
L1w
2 −
+
+Sw
+L4w
2
2 +
v a (v − w1 )b (v − w2 )c (1 − v)d dv
and
e2πia − 1
R1− (w1 , w2 ) = 2πi(a+b+c)
e
−1
+e
2πi(a+b+c+d)
Z
2πi(a+b+c)
+e
+
Sw
+L1w
2
L2w
2 −
Z
2πid
L4w
2 +
a
Z
−e
−
−Sw
−L2w
2
b
2 −
c
2 −
−
+Sw
+L3w+
2
Z
L1w
2 −
+
+Sw
+L4w
2
−
Z
L3w
2 +
2 +
d
× v (v − w1 ) (v − w2 ) (1 − v) dv,
where the principal branch is used for all powers. A computation gives
Q1− (w1 , w2 ) − R1− (w1 , w2 ) =
2iπc
× (1 − e
Z
)
e2πi(a+b) (e2πia − 1)(e2πid − 1)
(e2πi(a+b) − 1)(e2πi(a+b+c) − 1)
2πi(a+b+c)
L1w
+e
2iπc
(1 − e
2 −
× v a (v − w1 )b (v − w2 )c (1 − v)d dv,
Z
)
L2w
+e
2πic
2 −
0 < w1 < w2 < 1.
71
2πi(a+b)
(e
− 1)
Z
+
−
Sw
+Sw
2
2
Using the identity
(
c
(v − w2 ) =
eiπc (w2 − v)c ,
e−iπc (w2 − v)c ,
v ∈ Sw+2 ∪ L1w2 − ,
v ∈ Sw−2 ∪ L2w2 − ,
we can write this as
e2πi(a+b) (e2πia − 1)(e2πid − 1)eiπc
Q1− (w1 , w2 ) − R1− (w1 , w2 ) =
(1 − e2iπc )
(e2πi(a+b) − 1)(e2πi(a+b+c) − 1)
+ e2πi(a+b) (1 − e2iπc )
Z
L2w
+e2πic (e2πi(a+b) − 1)
2 −
× v a (v − w1 )b (w2 − v)c (1 − v)d dv,
Z
+
Sw
2
+(e2πi(a+b) − 1)
Z
Z
L1w
2 −
−
Sw
2
0 < w1 < w2 < 1.
That is,
Q1− (w1 , w2 ) − R1− (w1 , w2 ) =
×
Z (0+,w2 +,0−,w2 −)
w2 −
e2πi(a+b) (e2πia − 1)(e2πid − 1)eiπc
(e2πi(a+b) − 1)(e2πi(a+b+c) − 1)
v a (v − w1 )b (w2 − v)c (1 − v)d dv,
0 < w1 < w2 < 1,
where w1 lies inside the contour in the same component as 0. Applying the change of
variables s = wv2 , which maps the interval (0, w2 ) to the interval (0, 1), we obtain
Q1− (w1 , w2 ) − R1− (w1 , w2 ) =
×
Z (0+,1+,0−,1−)
e2πi(a+b) (e2πia − 1)(e2πid − 1)eiπc a+c+1
w
(e2πi(a+b) − 1)(e2πi(a+b+c) − 1) 2
sa (sw2 − w1 )b (1 − s)c (1 − sw2 )d ds,
0 < w1 < w2 < 1,
A
where A ∈ (0, 1) is so large that Aw2 − w1 > 0. Equation (10.28) follows.
The identity (10.26) can be used to determine the asymptotics of F (w1 , w2 ) as w1 → 0
and w2 → 0 to all orders.
10.6
Asymptotics of F as w1 → 0 and w2 → 1
We finally consider the asymptotics of F (w1 , w2 ) in the sector where w1 → 0 and w2 → 1.
Assuming that a + b, c + d ∈
/ Z, we define two functions Q̃1 : D1 → C and T1 : D0 → C as
follows. We define Q̃1 by
e2πia − 1
Q̃1 (w1 , w2 ) = 2πi(a+b)
e
−1
Z (0+,1+,0−,1−)
v a (v − w1 )b (w2 − v)c (1 − v)d dv,
(w1 , w2 ) ∈ D1 ,
A
where A ∈ (0, 1), w1 lies inside the contour in the same component as 0, and w2 lies
outside the contour. Then
Q1 (w1 , w2 ) = ρ(w2 )Q̃1 (w1 , w2 ),
72
(w1 , w2 ) ∈ D0 ∩ D1 ,
where ρ is given by (10.8). For 0 < Re w1 < Re w2 < 1 with Im w1 < 0 and Im w2 > 0, we
define T1 (w1 , w2 ) by
T1 (w1 , w2 ) =
(e2πia − 1)(e2πid − 1)
(e2πi(a+b) − 1)(e2πi(c+d) − 1)
Z (0+,1+,0−,1−)
v a (v − w1 )b (w2 − v)c (1 − v)d dv,
A
where w1 lies inside the contour in the same component as 0, w2 lies inside the contour in
the same component as 1, and we assume that Re w1 < A < Re w2 . We then use analytic
continuation to extend T1 to all of D1 .
Lemma 10.4. Suppose a + b, c + d ∈
/ Z. Then
Q̃1 (w1 , w2 ) = T1 (w1 , w2 ) + (w2 − 1)c+d+1 P2 (w1 , w2 ),
(w1 , w2 ) ∈ D1 .
(10.29)
Proof. By analyticity, it is enough to show that
Q̃1∗ (w1 , w2 ) = T1∗ (w1 , w2 ) + |1 − w2 |c+d+1 eiπ(c+d+1) P2∗ (w1 , w2 ),
0 < w1 < w2 < 1,
(10.30)
where Q̃1∗ (w1 , w2 ) := Q̃1 (w1 − i0, w2 + i0) etc.
Let 0 < w1 < w2 < 1 and let 0 < < 21 min{w1 , w2 − w1 , 1 − w2 }. Then
e2πia − 1
Q̃1∗ (w1 , w2 ) = 2πi(a+b)
e
−1
−e
2πi(a+b+d)
Z
L1w
+e
Z
2πi(a+b)
L2w
2 −
Z
2πid
−L1w
2 −
2 −
−
Z
L3w
2 +
−
Z
−
Sw
2
−
+Sw
+L3w
2
Z
+e
−
+L2w
Sw
2
2 −
−
+Sw
2
+e
2πi(a+b+c+d)
L4w
2 +
−e
Z
2 +
2πi(c+d)
Z
L4w
2 +
v a (v − w1 )b (w2 − v)c (1 − v)d dv,
and
T1∗ (w1 , w2 ) =
(e2πia − 1)(e2πid − 1)
(e2πi(a+b) − 1)(e2πi(c+d) − 1)
2πi(a+b+c+d)
+e
Z
L1w
2 +
Z
L2w
2 −
Z
L4w
+e2πi(a+b)
−e
+
+Sw
−L2w
2
2 −
2πi(c+d)
−
+Sw
+L3w
2
2 +
Z
L1w
2 −
Z
2 −
+
+Sw
+L4w
2
+
2 +
−L3w
2 +
−
−Sw
2
× v a (v − w1 )b (w2 − v)c (1 − v)d dv.
It follows that
e2πia − 1
(1 − e−2iπc )
Q̃1∗ (w1 , w2 ) − T1∗ (w1 , w2 ) =
1 − e−2iπ(c+d)
+e
2πid
2iπc
(e
− 1)
Z
Z
L3w
2 +
2πid
L4w
+(1 − e
2 +
× v a (v − w1 )b (w2 − v)c (1 − v)d dv,
73
Z
)
+
−
Sw
+Sw
2
2
0 < w1 < w2 < 1.
Using the identity
(
c
(w2 − v) =
e−iπc (v − w2 )c ,
eiπc (v − w2 )c ,
v ∈ Sw+2 ∪ L4w2 + ,
v ∈ Sw−2 ∪ L3w2 + ,
we can write this as
(e2πia − 1)e−iπc
(e2iπc − 1)
1 − e−2iπ(c+d)
Q̃1∗ (w1 , w2 ) − T1∗ (w1 , w2 ) =
+ (1 − e2πid )
Z
(e2πia − 1)e−iπc
=
1 − e−2iπ(c+d)
+
Sw
2
+e2iπc (1 − e2πid )
Z (w2 +,1+,w2 −,1−)
Z
−
Sw
2
Z
L3w
+e2πid (e2iπc − 1)
2 +
Z
L4w
2 +
v a (v − w1 )b (v − w2 )c (1 − v)d dv
v a (v − w1 )b (v − w2 )c (1 − v)d dv,
0 < w1 < w2 < 1,
w2 +
where w1 lies exterior to the Pochhammer contour. Performing the change of variables
2
s = v−w
1−w2 , which maps the interval (w2 , 1) to the interval (0, 1), we obtain
Q̃1∗ (w1 , w2 ) − T1∗ (w1 , w2 ) =
×
Z (0+,1+,0−,1−)
(e2πia − 1)e−iπc
(1 − w2 )c+d+1
1 − e−2iπ(c+d)
(w2 + s(1 − w2 ))a (w2 + s(1 − w2 ) − w1 )b sc (1 − s)d ds,
0 < w1 < w2 < 1,
A
where A ∈ (0, 1). Since
(e2πia − 1)e−iπc
eiπ(a+d) sin(aπ)
,
=
sin(π(d + c))
1 − e−2iπ(c+d)
equation (10.30) follows from (10.16).
Recalling (10.22), equation (10.29) yields
F (w1 , w2 ) = ρ(w2 ) T1 (w1 , w2 ) + (w2 − 1)c+d+1 P2 (w1 , w2 ) + w1a+b+1 Q2 (w1 , w2 ) (10.31)
for (w1 , w2 ) ∈ D0 ∩ D1 . The identity (10.31) can be used to find the asymptotics of
F (w1 , w2 ) as w1 → 0 and w2 → 1.
10.7
Some basic estimates
As explained above, the identities (10.17), (10.22), (10.26), and (10.31) can be used to
determine the asymptotics to all orders of F (w1 , w2 ) as one or both of the points w1 , w2
approach 0 or 1. However, for the purposes of establishing Lemma 6.2 (the result relevant
for the SLEκ (2) Green’s function), we only need some leading and subleading estimates
on F . These estimates can be derived from the identities (10.17), (10.22), (10.26), and
(10.31) together with a number of bounds on the functions P1 , Qj , Rj , T1 , F . We collect
the required bounds in the next lemma.
If w ∈ C and A is a subset of C, we write dist(w, A) for the Euclidean distance from
w to A; we write dist(w, A ∪ {∞}) > to indicate that dist(w, A) > and |w| < 1/.
74
Lemma 10.5. Suppose a, b > 0 and c, d 6 0 satisfy a, d, a + b, c + d, a + b + c ∈
/ Z. Let
> 0. Then the following estimates hold:
(a) |P1 (w1 , w2 )| 6 C and |P1 (w1 , w2 )−P1 (w1 , 1)| 6 C|w2 −1| uniformly for all (w1 , w2 ) ∈
C2 such that dist(w1 , {0, 1, ∞}) > and |w2 − 1| < 1 − .
(b) |Q1 (w1 , w2 )| 6 C|w2 |c uniformly for all (w1 , w2 ) ∈ C2 such that |w1 | < 1 − and
dist(w2 , {0, 1}) > .
(c) |Q2 (w1 , w2 )| 6 C|w1 |c uniformly for all (w1 , w2 ) ∈ C2 \ {(0, 0)} such that |w1 | < 1 − w2
and dist( w
, {0, 1}) > .
1
(d) |R1 (w1 , w2 )| 6 C and |R1 (w1 , w2 ) − R1 (0, 0)| 6 C(|w1 | + |w2 |) uniformly for all
(w1 , w2 ) ∈ C2 such that |w1 | < 1 − and |w2 | < 1 − .
w2
(e) |R2 (w1 , w2 )| 6 C|w2 |b uniformly for all (w1 , w2 ) ∈ C2 \{(0, 0)} such that dist( w
, {0, 1}) >
1
and |w2 | < 1 − .
(f ) |T1 (w1 , w2 )| 6 C uniformly for all (w1 , w2 ) ∈ C2 such that |w1 | < 1− and dist(w2 , {0, ∞}) >
.
(g) |F (w1 , w2 )| 6 C|w2 |c uniformly for all (w1 , w2 ) ∈ C2 such that dist(w1 , {0, 1, ∞}) > and |w2 | > 1 + .
Proof. The estimates follow easily from the definitions of the functions P1 , Qj , Rj , T1 , F .
Remark 10.6. If (w1 , w2 ) lies on a branch cut, the bounds in Lemma 10.5 should be
interpreted as saying that both the left and right boundary values obey the bounds.
11
Differential equations
In this section we briefly discuss our results from the point of view of differential equations.
We know from the proofs of Theorem 2.1 and Theorem 2.6 that the observables are
smooth (so Itô’s formula can be applied), and this makes it easy to write down differential
equations for them using standard methods. We consider only the case of the Green’s
function for a system of commuting SLEs started from (ξ 1 , ξ 2 ) with ξ 1 < ξ 2 . We write
the Green’s function as
lim d−2 P (Υ∞ (z) 6 ) = G(x, y, ξ 1 , ξ 2 ),
→0
(z = x + iy).
(11.1)
Scale invariance of SLE implies that G is scale invariant in the sense that
G(λx, λy, λξ 1 , λξ 2 ) = λd−2 G(x, y, ξ 1 , ξ 2 ),
λ > 0,
and we can write
G(x, y, ξ 1 , ξ 2 ) = y d−2 H(x, y, ξ 1 , ξ 2 ),
(11.2)
where the function H is homogeneous:
H(λx, λy, λξ 1 , λξ 2 ) = H(x, y, ξ 1 , ξ 2 ),
75
λ > 0.
(11.3)
Proposition 11.1. Let κ 6 4 and write G(x, y, ξ 1 , ξ 2 ) for the Green’s function for a
system of commuting SLEκ . Then the function H defined by the relation (11.2) satisfies
the following two PDEs:
2a(x − ξ j )
2ay
2a
2a
∂x H − 2
∂y H + 1
∂ 1H + 2
∂ 2H
+ (x − ξ j )2
y + (x − ξ j )2
ξ − ξ2 ξ
ξ − ξ1 ξ
βy 2
+ 2
H = 0,
j = 1, 2,
(11.4)
(y + (x − ξ j )2 )2
∂ξ2j H +
y2
where (x, y, ξ 1 , ξ 2 ) ∈ R4 with y > 0 and ξ 1 < ξ 2 .
Proof. This follows from the smoothness of G using Itô’s formula together with Loewner’s
equation. We omit the details.
Remark 11.2. Schramm’s probability satisfies the same PDEs without the last term on
the left-hand side.
In addition to the homogeneity property (11.3), the function H is also translation
invariant in the x-direction, i.e.,
H(x, y, ξ 1 , ξ 2 ) = H(x + λ, y, ξ 1 + λ, ξ 2 + λ),
λ ∈ R.
(11.5)
Using the two symmetries (11.3) and (11.5), we can reduce the PDEs in (11.4) from four to
two dimensions. In fact, one can prove that the function h(x, ξ) := H(x, 1, −ξ, ξ) satisfies
an elliptic PDE. Note, however, that we are crucially using the smoothness to draw this
conclusion. Working from (11.1) we can only derive the PDEs formally without additional
arguments establishing smoothness.
A
A.1
Estimates for Schramm’s formula
Properties of the function J(z, ξ)
Recall that we defined J(z, ξ) in (5.2) by
Z z
J(z, ξ) =
α
α
(u − z)α (u − z̄)α−2 u− 2 (u − ξ)− 2 du,
z̄
z ∈ H,
ξ > 0,
where the contour from z̄ to z passes to the right of ξ, see Figure 1.
Lemma A.1. The function J(z, ξ) defined in (5.2) is a well-defined smooth function of
(z, ξ) ∈ H × (0, ∞).
Proof. Since α > 1, the integral defining J(z, ξ) is convergent for each z ∈ H and each
ξ > 0. To prove the smoothness of J, we first assume that α > 1 is an integer. In this
case the integral in (5.2) can be computed explicitly in terms of logarithms and powers
of z, z̄, z − ξ, and z̄ − ξ (see Section 5.2 for the case α = 2). Hence J(z, ξ) is smooth for
(z, ξ) ∈ H × (0, ∞).
76
Im z
z
l1
l3
0
Re z
A
ξ
l2
z̄
l4
Figure 10. The integration contour in (A.1) is the composition of the four loops lj ,
j = 1, . . . , 4, based at the point A > ξ.
Assume α > 1 is not an integer. Then, fixing a basepoint A > ξ, we can rewrite the
expression (5.2) for J(z, ξ) as
J(z, ξ) =
1
(1 − e2iπα )2
Z (z+,z̄+,z−,z̄−)
α
α
(u − z)α (u − z̄)α−2 u− 2 (u − ξ)− 2 du,
A
z ∈ H,
ξ > 0,
(A.1)
where the integration contour is the composition of four loops {lj }41 based at A (see
Figure 10) and the integrand is evaluated using analytic continuation along the contour.
More precisely, the loop l1 encircles z once in the counterclockwise direction, l2 encircles
z̄ once in the counterclockwise direction, l3 encircles z once in the clockwise direction,
and l4 encircles z̄ once in the clockwise direction. On the first half of l1 , the principal
branch is used, but as the contour l1 encircles z in the counterclockwise direction, the
power (u − z)α in the integrand picks up an additional factor of e2iπα with respect to
the principal branch; then, as l2 encircles z̄ in the counterclockwise direction, the power
(u − z̄)α−2 in the integrand picks up the factor e2iπ(α−2) and so on. Collapsing the contour
onto a single path from z̄ to z and collecting the exponential factors, we see that (A.1)
reduces to (5.2). Since the contour in (A.1) avoids the branch points, the integral in
(A.1) can be differentiated an unlimited number of times with respect to z, z̄, and ξ. This
completes the proof of the lemma.
Lemma A.2. The function J(z, ξ) defined in (5.2) satisfies the following estimates:
|J(z, ξ)| 6 C|z − ξ|α−1 ,
|J(z, ξ)| 6 C|x|
−α
2
|x − ξ|
−α
2α−1
2
y
,
77
z ∈ H,
ξ > 0,
x > ξ,
y > 0,
(A.2a)
ξ > 0.
(A.2b)
|Re J(z, ξ)| 6 Cy|z|α−2 ,
|z| > 2ξ,
z ∈ H,
ξ > 0,
(A.2c)
where z = x + iy.
Proof. To prove (A.2a), we let z = ξ + reiθ and choose the following parametrization of
the integration contour in (5.2) (see Figure 11):
u = ξ + reiϕ ,
−θ 6 ϕ 6 θ.
(A.3)
This yields after simplification
iθ
J(ξ + re , ξ) = ir
3α
−1
2
Z θ
α
α
(eiϕ − eiθ )α (eiϕ − e−iθ )α−2 (ξ + reiϕ )− 2 eiϕ(1− 2 ) dϕ,
−θ
θ ∈ (0, π),
r > 0,
ξ > 0.
(A.4)
It follows that
iθ
|J(ξ + re , ξ)| 6 r
3α
−1
2
Z θ
α
|eiϕ − eiθ |α |eiϕ − e−iθ |α−2 |ξ + reiϕ |− 2 dϕ.
−θ
Since
|reiϕ + r|
,
2
|ξ + reiϕ | >
for all r > 0, ϕ ∈ (−π, π), and ξ > 0, we obtain the estimate
|J(ξ + reiθ , ξ)| 6 Crα−1
Z θ
α
|eiϕ − eiθ |α |eiϕ − e−iθ |α−2 |eiϕ + 1|− 2 dϕ.
−θ
The integral remains bounded as θ ↑ π, because
|J(ξ + reiθ , ξ)| 6 Crα−1 ,
3α
2
r > 0,
− 2 > −1. Thus we arrive at
θ ∈ (0, π),
ξ > 0,
which is (A.2a).
To prove (A.2b), we let z = x + iy in (5.2) and use the parametrization u = x + is,
−y 6 s 6 y, of the contour from z̄ to z. Assuming that x > ξ, this yields
Z y
J(z, ξ) =
α
α
(is − iy)α (is + iy)α−2 (x + is)− 2 (x − ξ + is)− 2 ids.
−y
It follows that
|J(z, ξ)| 6
Z y
α
α
|s − y|α |s + y|α−2 |x + is|− 2 |x − ξ + is|− 2 ds
−y
Z y
α
α
−α
2
−y
−α
2α−1
2
6 |x|− 2 |x − ξ|− 2 (2y)α
6 C|x|
|x − ξ|
y
,
This proves (A.2b).
78
|s + y|α−2 ds
x > ξ,
y > 0,
ξ > 0.
(A.5)
Im z
z
r
θ
Re z
0
ξ
z̄
Figure 11. The contour from z̄ = ξ + re−iθ to z = ξ + reiθ defined in equation (A.3).
To prove (A.2c), we note that if f (u) is an analytic function, then
Z z
f (u)du = −
Z z
f (ū)du.
(A.6)
z̄
z̄
Hence
Z z
2Re J(z, ξ) =
α
α
(u − z)α (u − z̄)α−2 u− 2 (u − ξ)− 2 du
z̄
−
Z z
α
α
(u − z)α−2 (u − z̄)α u− 2 (u − ξ)− 2 du.
z̄
Since
(u − z)2 − (u − z̄)2 = −4iy(u − x),
this can be written as
Re J(z, ξ) = −2iy
Z z
α
α
(u − z)α−2 (u − z̄)α−2 u− 2 (u − ξ)− 2 (u − x)du.
z̄
Assuming that z = reiθ satisfies |z| > ξ and adopting the parametrization
u = reiϕ ,
−θ 6 ϕ 6 θ,
of the integration contour from z̄ to z, we arrive at
Re J(reiθ , ξ) = 2yr
3α
−2
2
Z θ
α
(eiϕ − eiθ )α−2 (eiϕ − e−iθ )α−2 (reiϕ − ξ)− 2
−θ
α
× (eiϕ − cos θ)eiϕ(1− 2 ) dϕ.
In view of the estimate
|reiϕ − ξ| >
79
r
2
(A.7)
Im z
x
Re z
0
ξ
Lx
Figure 12. The integration contour Lx in (A.8) is a loop which encloses ξ in the counterclockwise direction.
valid for all r > 2ξ, ϕ ∈ (0, π), and ξ > 0, this implies
iθ
|Re J(re , ξ)| 6 Cyr
α−2
Z θ
|eiϕ − eiθ |α−2 |eiϕ − e−iθ |α−2 |eiϕ − cos θ|dϕ
−θ
r > 2ξ,
θ ∈ (0, π),
ξ > 0.
Since 2α − 3 > −1, the integral remains bounded as θ ↑ π. This proves (A.2c).
We next extend the function J(z, ξ) continuously to all of H̄ × (0, ∞). As suggested
by (5.2), we define J(x, ξ) for x ∈ R by
(
J(x, ξ) =
0,
R
Lx (u
−
α
x)2α−2 u− 2 (u
− ξ)
−α
2
x > ξ,
du, x < ξ,
(A.8)
where the integration contour Lx is a loop starting and ending at x which avoids the
branch cut along (−∞, ξ) and which encloses ξ in the counterclockwise direction, see
Figure 12.
Lemma A.3. For each ξ > 0, the function J(z, ξ) defined by (5.2) and (A.8) is a continuous function of z ∈ H̄.
Proof. Fix ξ > 0. Since α > 1, the integral in (A.8) converges for every x < ξ (including
x = 0). Equation (A.2b) implies that z 7→ J(z, ξ) is continuous at each point in (ξ, ∞).
Moreover, equation (A.2a) implies that J is continuous at z = ξ.
We next show that J is continuous at each point in (−∞, 0) ∪ (0, ξ). Letting s = πθ ϕ
and simplifying, we can write (A.4) as
Z π
iθ
J(ξ + re , ξ) =
−π
80
gr,θ (s)ds,
where
gr,θ (s) = ir
3α
−1
2
α iθs
α−2
iθs − α iθs
α
θ iθs
2
ξ + re π
e π (1− 2 ) .
e π − eiθ e π − e−iθ
π
Let > 0. Then
|gr,θ (s)| 6 C e
iθs
π
α iθs
α−2
− eiθ e π − e−iθ α−2 θs
θs α−2 , θ +
6 C2 max θ + − 2π π
π
α−2
α−2 α
6 C max |π + s|
, |π − s|
,
s ∈ (−π, π),
for all r ∈ (, −1 ) with |r − ξ| > and all θ ∈ [ π2 , π]. Since |π ± s|α−2 ∈ L1 ([−π, π]), this
shows that there exists a function G(s) in L1 ([−π, π]) such that
|gr,θ (s)| 6 G(s),
s ∈ (−π, π),
(A.9)
for all r ∈ (, −1 ) with |r − ξ| > and all θ ∈ [ π2 , π]. Since > 0 was arbitrary, if the
point x0 = ξ + r0 eiπ belongs to (−∞, 0) ∪ (0, ξ), dominated convergence gives
Z π
Z π
lim J(z, ξ) = lim
z→x0
θ↑π −π
r→r0
gr,θ (s)ds =
−π
gr0 ,π (s)ds = J(x0 , ξ).
This shows that J(·, ξ) is continuous at each point in (−∞, 0) ∪ (0, ξ).
It only remains to show that J(·, ξ) : H̄ → C is continuous at z = 0. To prove this, let
c = 2ξ and let z = reiθ with θ ∈ [0, π] and r ∈ (0, c). Let γ denote the contour from z̄ to
z used in the definition (5.2) of J(z, ξ). We write γ as the union of five subcontours as
follows (see Figure 13):
γ = γ1 + γ2 + γ3 + γ4 + γ5 ,
where
γ1 : {reiϕ | − θ 6 ϕ 6 0},
γ2 : {u − i0 | r 6 u 6 ξ − c},
γ3 : {ξ + ceiϕ | − π 6 ϕ 6 π},
γ4 : {u + i0 | r 6 u 6 ξ − c},
iϕ
γ5 : {re | 0 6 ϕ 6 θ}.
(A.10)
Then
J(z, ξ) =
5
X
Jj (z),
j=1
where
Z
Jj (z) =
α
α
(u − z)α (u − z̄)α−2 u− 2 (u − ξ)− 2 du,
j = 1, . . . , 5.
γj
We claim that
|Jj (z)| 6 Cr
3α
−1
2
,
|z| < c,
81
z ∈ H,
j = 1, 5.
(A.11)
Im z
γ5
z
γ4
Re z
ξ
γ2
z̄
γ1
γ3
Figure 13. The integration contour γ is the composition of the five subcontours γj , j =
1, . . . , 5, defined in (A.10).
Indeed, let us consider the case of J1 (z). Since |u − ξ| > c for u ∈ γ1 , we have
|J1 (z)| 6
Z
α
α
|u − z|α |u − z̄|α−2 r− 2 c− 2 |du|.
γ1
Moreover, the inequalities
|u − z̄| 6 |u − z| 6 2r
are valid for u ∈ γ1 . Hence, if α > 2, then
|J1 (z)| 6 C
Z
|u − z|
2α−2 − α
2
r
γ1
|du| 6 C
Z
r
3α
−2
2
γ1
|du| 6 Cr
3α
−1
2
.
On the other hand, if 1 < α < 2, then
|J1 (z)| 6 C
Z
α
(2r) |u − z̄|
α−2 − α
2
r
γ1
|du| 6 Cr
α
2
Z θ
|re−iϕ − re−iθ |α−2 rdϕ
0
α−2
Z θ
2
3α
3α
3α
θα−1
−1
2
6 Cr
6 Cr 2 −1 .
dϕ 6 Cr 2 −1
π (θ − ϕ)
α−1
0
This proves (A.11) for J1 (z); the proof for J5 (z) is similar.
We next show that
lim Jj (z) = Jj (0),
z→0
j = 2, 4.
To establish (A.12), we let u = r + s and write
Z c
J2 (z) =
fr,θ (s)ds,
0
82
(A.12)
where
α
α
fr,θ (s) = χ[0,c−r] (s)(s + r − reiθ )α (s + r − re−iθ )α−2 (s + r)− 2 |s + r − ξ|− 2 e
αiπ
2
and χ[0,c−r] denotes the characteristic function of the interval [0, c − r]. We will show that
there exists a function F (s) in L1 ((0, c)) such that
|fr,θ (s)| 6 F (s),
s ∈ (0, c),
(A.13)
for all r ∈ (0, c) and all θ ∈ [0, π]. Dominated convergence then gives
Z c
Z c
lim J2 (z) = lim
z→0
r→0 0
f0,θ (s)ds = J2 (0),
fr,θ (s)ds =
0
showing that J2 (z) satisfies (A.12).
In order to prove (A.13), we note that
|s + r − reiθ | = |s + r − re−iθ | and
|s + r − ξ| > c
for s ∈ (0, c − r). This gives
α
α
|fr,θ (s)| 6 |s + r − reiθ |2α−2 (s + r)− 2 c− 2 .
Using the inequalities
|s + r − reiθ | 6 s + 2r,
s+r >
s + 2r
,
2
we find
|fr,θ (s)| 6 |s + 2r|
3α
−2
2
α
s ∈ (0, c),
(2/c) 2 ,
Since
|s + 2r|
3α
−2
2
3α
(3c) 2 −2 ,
3α
s 2 −2 ,
(
6
r ∈ (0, c),
θ ∈ [0, π].
α > 43 ,
1 < α < 43 ,
we deduce that (A.13) holds with
α
F (s) = (2/c) 2 max((3c)
3α
−2
2
,s
3α
−2
2
),
0 < s < c.
This proves (A.12) for j = 2; the proof when j = 4 is similar.
Finally, since the integration contour is independent of z, it is easy to see that
lim J3 (z) = J3 (0).
z→0
(A.14)
Since J(z, ξ) = 5j=1 Jj (z), the continuity of J(z, ξ) at z = 0 follows from equations
(A.11), (A.12), and (A.14). This completes the proof of the lemma.
P
Lemma A.4. For each ξ > 0, the partial derivatives ∂x J(z, ξ) and ∂y J(z, ξ) have continuous extensions to H̄ \ {0, ξ}.
83
Proof. Fix ξ > 0. Defining W (u) by
α
α
W (u) = u− 2 (u − ξ)− 2 ,
(A.15)
we can write
Z z
J(z, ξ) =
(u − z)α (u − z̄)α−2 W (u)du.
z̄
An integration by parts gives
z
α
(u − z)α−1 (u − z̄)α−1 W (u)du
α − 1 z̄
Z z
1
−
(u − z)α (u − z̄)α−1 W 0 (u)du.
α − 1 z̄
Z
J(z, ξ) = −
(A.16)
Differentiating with respect to z and z̄, we find
Z z
Jz (z, ξ) = α
(u − z)α−2 (u − z̄)α−1 W (u)du
z̄
+
α
α−1
Z z
(u − z)α−1 (u − z̄)α−1 W 0 (u)du
z̄
and
Z z
Jz̄ (z, ξ) = α
(u − z)α−1 (u − z̄)α−2 W (u)du
Zz̄ z
+
(u − z)α (u − z̄)α−2 W 0 (u)du.
z̄
Since α > 1, these expressions for Jz and Jz̄ are well-defined for each z ∈ R \ {0, ξ}.
Repeating the above arguments that led to the continuity of J(z, ξ) at each point z ∈
R \ {0, ξ}, we infer that this provides continuous extensions of Jz and Jz̄ to H̄ \ {0, ξ}.
Lemma A.5. For each fixed ξ > 0, the function J(z, ξ) satisfies
x ∈ R,
Re J(x, ξ) = 0,
ξ > 0.
(A.17)
Moreover,
(
J(x + iy, ξ) = O(1),
Re J(x + iy, ξ) = O(y),
y ↓ 0,
x ∈ R \ {0, ξ},
(A.18a)
and
J(x + iy, ξ) = O(y 2α−1 ),
y ↓ 0,
x > ξ,
(A.18b)
where the error terms are uniform with respect to x in compact subsets of R \ {0, ξ}.
Proof. Equation (A.17) follows by letting y → 0 in (A.7). The asymptotic formulas
(A.18a) are then a direct consequence of Lemma A.3 and Lemma A.4. Equation (A.18b)
follows from (A.2b).
84
A.2
Existence and regularity of P
This section studies the integral in (2.5) predicted to equal Schramm’s formula.
Lemma A.6. The function M(z, ξ) defined in (2.4) satisfies the following estimates:
|M(z, ξ)| 6 Cy α−2 |z|1−α ,
|M(z, ξ)| 6 Cy
3α−3
1−α
|z|
z ∈ H,
1−α
|z − ξ|
ξ > 0,
−α
2
|x|
(A.19a)
−α
2
|x − ξ|
,
x > ξ,
y > 0,
ξ > 0,
(A.19b)
z ∈ H,
ξ > 0,
(A.19c)
h
|Re M(z, ξ)| 6 Cy α−1 |z|−α |z − ξ|−α (|x||x − ξ| + y 2 )|z|α−2
i
+ (|x| + |ξ − x|)|z − ξ|α−1 ,
|z| > 2ξ,
where z = x + iy. In particular, for each fixed ξ > 0,
M(z, ξ) = O(|x|1−α ),
−α
Re M(z, ξ) = O(|x|
x → ±∞,
x → ±∞,
),
y > 0,
(A.20a)
y > 0,
(A.20b)
where the error terms are uniform with respect to y in compact subsets of (0, ∞).
Proof. The estimates (A.19a) and (A.19b) follow immediately from the definition of M
together with the estimates (A.2a) and (A.2b) of Lemma A.2. The estimate (A.19c)
follows by applying (A.2a) and (A.2c) to the identity
Re M(z, ξ) = y α−2 |z|−α |z − ξ|−α Re z̄(z̄ − ξ)J(z, ξ)
= y α−2 |z|−α |z − ξ|−α (x2 − xξ − y 2 )Re J(z, ξ) − y(ξ − 2x)Im J(z, ξ) .
(A.21)
Finally, the asymptotic equations in (A.20) are an immediate consequence of (A.19a) and
(A.19c).
Lemma A.7. The function M = M1 + iM2 satisfies
∂y M1 (z, ξ) = −∂x M2 (z, ξ),
z ∈ H,
ξ > 0.
Proof. Since the statement only involves derivatives with respect to x and y, we can
¯ = 0. In terms of the function
assume that ξ > 0 is fixed. We need to prove that Im ∂M
W (u) defined in (A.15) we can write
α
α
α
α
M(z, ξ) = y α−2 z − 2 (z − ξ)− 2 z̄ 1− 2 (z̄ − ξ)1− 2
Z z
(u − z)α (u − z̄)α−2 W (u)du.
z̄
An integration by parts gives
α
α
α
α
M(z, ξ) = −y α−2 z − 2 (z − ξ)− 2 z̄ 1− 2 (z̄ − ξ)1− 2
Z z
× α
(u − z)
α−1
(u − z̄)
α−1
Z z
W (u)du +
z̄
z̄
85
1
α−1
α
α−1
(u − z) (u − z̄)
0
W (u)du .
(A.22)
Differentiating with respect to z̄, we find
¯
∂M(z,
ξ) =
i α−2 1−
+
2 y
z̄
α
α
α
2
1 − α2
M(z, ξ)
z̄ − ξ
+
α
α
+ y α−2 z − 2 (z − ξ)− 2 z̄ 1− 2 (z̄ − ξ)1− 2
Z z
× α
(u − z)
α−1
(u − z̄)
α−2
Z z
W (u)du +
α
α−2
(u − z) (u − z̄)
0
W (u)du .
(A.23)
z̄
z̄
The integrals in (A.23) are convergent at the endpoints z and z̄ because α > 1. Substituting the expression (A.22) for M into (A.23) and simplifying, we obtain
¯
∂M(z,
ξ) = R(z, ξ) − i(α − 2) x(x − ξ) + y 2
Z z
× α
(u − z)
α−1
(u − z̄)
α−1
Z z
W (u)du +
α
α−1
α
α−2
(u − z) (u − z̄)
0
W (u)du
z̄
z̄
+ 2(α − 1)yz̄(z̄ − ξ)
Z z
× α
(u − z)
α−1
(u − z̄)
α−2
Z z
W (u)du +
z̄
(u − z) (u − z̄)
0
W (u)du
, (A.24)
z̄
where the real-valued function R(z, ξ) is given by
R(z, ξ) =
1
y α−3 |z|−α |z − ξ|−α .
2(α − 1)
¯ = 0 it is enough to show that
To establish the identity Im ∂M
¯
¯ − ∂M
∂M
= 0.
R
(A.25)
Using (A.6) and (A.24), we can write the left-hand side of (A.25) as
− i(α − 2) x(x − ξ) + y 2
Z z
× α
(u − z)
α−1
(u − z̄)
α−1
Z z
W (u)du +
α
α−1
α
α−2
(u − z) (u − z̄)
0
W (u)du
z̄
z̄
+ 2(α − 1)yz̄(z̄ − ξ)
Z z
× α
α−1
(u − z)
α−2
(u − z̄)
Z z
W (u)du +
z̄
W (u)du
z̄
− i(α − 2) x(x − ξ) + y 2
×
(u − z) (u − z̄)
0
−α
Z z
(u − z)
α−1
(u − z̄)
α−1
W (u)du −
z̄
Z z
(u − z)
α−1
(u − z)
α−2
α
0
α
0
(u − z̄) W (u)du
z̄
− 2(α − 1)yz(z − ξ)
×
−α
Z z
(u − z)
α−2
(u − z̄)
α−1
W (u)du −
z̄
Z z
z̄
86
(u − z̄) W (u)du . (A.26)
The right-hand side of (A.26) involves eight integrals. Integrating by parts in the third,
fourth, and eighth of these integrals and using that the first and fifth integrals cancel, we
see that the expression in (A.26) can be written as
i(α − 2) x(x − ξ) + y 2
×
−
Z z
α
α−1
(u − z) (u − z̄)
Z z
0
W (u)du +
(u − z)
α−1
α
0
(u − z̄) W (u)du
z̄
z̄
+ 2(α − 1)yz̄(z̄ − ξ)
(u − z̄)α−1 0
W (u)du
α−1
z̄
z̄
Z z
Z z
α−1
α−1
α (u − z̄)
α−1 (u − z̄)
0
00
(u − z)
(u − z)
W (u)du −
W (u)du
−α
α−1
α−1
z̄
z̄
− 2(α − 1)yz(z − ξ)
×
−α
×
−α
Z z
Z z
(u − z)
α−2
(u − z̄)
α−1
W (u)du − α
Z z
(u − z)α−1
(u − z)α−2 (u − z̄)α−1 W (u)du
z̄
+α
Z z
(u − z)α−1
α−1
z̄
(u − z̄)α−1 W 0 (u)du +
Z z
(u − z)α−1
z̄
α−1
(u − z̄)α W 00 (u)du .
(A.27)
A long but straightforward computation shows that the expression in (A.27) equals
2αy
Z z
d
z̄
du
α
α
(u − z)α (u − z̄)α u− 2 (u − ξ)− 2
2x − ξ x − ξ
x
−
−
u−z
u
u−ξ
du.
Since α > 1, the fundamental theorem of calculus implies that the integral vanishes. This
proves (A.25) and completes the proof of the lemma.
Lemma A.8. The function P (z, ξ) defined in (2.5) is a well-defined smooth function of
(z, ξ) ∈ H × (0, ∞).
Proof. By Lemma A.1, M = M1 + iM2 is a smooth function of (z, ξ) ∈ H × (0, ∞).
Moreover, by equation (A.20), there exists an > 0 such that M1 (x + iy) = O(|x|−1− )
and M2 (x + iy) = O(|x|− ) as |x| → ∞ uniformly with respect to y in compact subsets of
(0, ∞). It follows that the integral inR the definition (2.5) of P converges. Furthermore,
Lemma A.7 shows that the integral (M1 dx − M2 dy) is independent of the path. We
infer that P (z, ξ) can be written as
1 z
P (z, ξ) = −
M1 (z 0 , ξ)dx0 − M2 (z 0 , ξ)dy 0
cα ∞
Z
1 z
=−
Re M(z 0 , ξ)dz 0 ,
z ∈ H,
cα ∞
Z
(A.28)
where z 0 = x0 + iy 0 and the contour of integration runs from ∞ + ic, c > 0, to z. Since
M(z, ξ) is a smooth function of (z, ξ) ∈ H × (0, ∞), so is P (z, ξ).
87
Lemma A.9. For each ξ > 0, the function M(z, ξ) satisfies
(
M(x + iy, ξ) = O(y α−2 ),
Re M(x + iy, ξ) = O(y α−1 ),
y ↓ 0,
x ∈ R \ {0, ξ},
(A.29)
and
M(x + iy, ξ) = O(y 3α−3 ),
y ↓ 0,
x > ξ,
where the error terms are uniform with respect to x in compact subsets of R \ {0, ξ}.
Proof. This follows from Lemma A.5 and the definition (2.4) of M.
Lemma A.10. For each ξ > 0, the function z 7→ P (z, ξ) defined in (2.5) has a continuous
extension to H̄ \ {0} which satisfies
(
−∞ < x < 0,
0 < x < ∞,
1,
0,
P (x, ξ) =
ξ > 0.
(A.30)
Proof. Fix ξ > 0. The expression (A.28) for P together with Lemma A.9 imply that there
exist real constants {Pj }31 such that the function
z 7→


P (z, ξ),




P ,
1

P2 ,





P3 ,
z
z
z
z
∈ H,
∈ (−∞, 0),
∈ (0, ξ),
∈ (ξ, ∞),
is continuous H̄ \ {0, ξ} → R. Letting z approach ∞ + i0 in (A.28), we deduce that P3 = 0.
It follows that
1
P (ξ + re , ξ) = − Re ir
cα
iθ
Z θ
iϕ
iϕ
M(ξ + re , ξ)e dϕ
0
for r > 0 and θ ∈ (0, π). In view of the estimate (A.19a), this yields
|P (ξ + reiθ , ξ)| 6 Cr
6 Cr
Z θ
|M(ξ + reiϕ , ξ)|dϕ
0
α−1
Z θ
(sinα−2 ϕ)|ξ + reiϕ |1−α dϕ,
r > 0,
ξ > 0.
(A.31)
0
Letting r ↓ 0, we infer that if we set P (ξ, ξ) = 0, then P (z, ξ) is continuous at z = ξ.
Moreover, taking the limit r ↓ 0 in (A.31) with θ = π, it follows that P2 = 0.
It only remains to prove that P3 = 1. This will follow if we can show that the
normalization constant cα defined in (2.6) satisfies
cα = −Re
Z
M(z, ξ)dz,
r > 0,
Sr
where Sr is a counterclockwise semicircle of radius r centered at 0:
Sr : reiϕ ,
0 6 ϕ 6 π.
88
(A.32)
Lemmas A.7 and A.9 show that the right-hand side of (A.32) is independent of r > 0. We
can therefore evaluate it in the limit as r ↓ 0. Recalling the definition of M, we write
Z
M(z, ξ)dz =
Z π
fr (ϕ)dϕ,
Sr
0
where
α
α
fr (ϕ) = i(sinα−2 ϕ)(reiϕ − ξ)− 2 (re−iϕ − ξ)1− 2 J(reiϕ , ξ).
The function J is bounded on each compact subset of H̄ by Lemma A.3. Hence fr (ϕ)
obeys the estimate
|fr (ϕ)| 6 C sinα−2 ϕ,
ϕ ∈ [0, π].
r < ξ/2,
Since the function sinα−2 ϕ belongs to L1 ((0, π)), dominated convergence yields
−Re
Z
M(z, ξ)dz = − Re lim
Sr
= − Re
Z π
r→0 0
Z π
fr (ϕ)dϕ = −Re
Z π
lim fr (ϕ)dϕ
0 r→0
α
α
i(sinα−2 ϕ)(−ξ + i0)− 2 (−ξ − i0)1− 2 J(0, ξ)dϕ
0
= − (Im J(0, ξ))ξ
1−α
Z π
sinα−2 ϕdϕ.
0
But setting r = ξ and θ = π in (A.4), we find that the value of J(z, ξ) at z = 0 is given by
J(0, ξ) = iξ α−1
Z π
(1 + eiϕ )
3α
−2
2
α
eiϕ(1− 2 ) dϕ,
ξ > 0.
−π
It follows that
−Re
Z π
Z
M(z, ξ)dz = −
iϕ
(1 + e )
3α
−2
2
iϕ(1− α
)
2
e
Z π
dϕ
sin
−π
Sr
α−2
ϕdϕ .
0
Since
Z π
iϕ
(1 + e )
3α
−2
2
e
iϕ(1− α
)
2
2πΓ
dϕ =
−π
Γ
3α
2
α
2
−1
Γ(α)
√
Z π
,
sin
α−2
ϕdϕ =
0
πΓ( α−1
2 )
,
Γ( α2 )
equation (A.32) follows.
Remark A.11. The proof of Lemma A.10 shows that the constant cα can be alternatively
expressed as
Z
∞
cα =
Re M(x, y, ξ)dx,
−∞
where the right-hand side is independent of the choice of y > 0 and ξ > 0.
Lemma A.12. We have
|P (z, ξ)| 6 C (arg z)α−1 ,
z ∈ H,
|P (z, ξ) − 1| 6 C (π − arg z)α−1 ,
89
ξ > 0,
z ∈ H,
(A.33a)
ξ > 0.
(A.33b)
Proof. Using that P (x, ξ) = 0 for x > 0, we can write
Z θ
r
P (re , ξ) = − Re
cα
iθ
M(reiϕ , ξ)ieiϕ dϕ,
r > 0,
θ ∈ (0, π),
ξ > 0.
0
The estimate (A.19a) now yields
iθ
|P (re , ξ)| 6 Cr
Z θ
0
iϕ
|M(re , ξ)|dϕ 6 C
Z θ
(sinα−2 ϕ)dϕ
0
6 C θα−1 ,
r > 0,
θ ∈ (0, π),
ξ > 0,
which is (A.33a). Similarly, since P (x, ξ) = 1 for x < 0, we can write
P (reiθ , ξ) = 1 +
r
Re
cα
Z π
M(reiϕ , ξ)ieiϕ dϕ,
r > 0,
θ ∈ (0, π),
ξ > 0.
θ
The estimate (A.19a) now yields
iθ
|P (re , ξ) − 1| 6 Cr
Z π
θ
iϕ
|M(re , ξ)|dϕ 6 C
Z π
(sinα−2 ϕ)dϕ
θ
6 C(π − θ)α−1 ,
r > 0,
θ ∈ (0, π),
ξ > 0,
which is (A.33b).
Consider a system of two commuting SLEs in H started from 0 and ξ > 0, respectively,
with sufficiently regular growth speeds λj (t) > 0. Write ξt1 and ξt2 for the Loewner driving
terms of the system and let gt denote the solution of (2.1) which uniformizes the system
at capacity t. Given z ∈ H, let Zt = gt (z) and let τ (z) denote the time that z is swallowed
by the system.
Lemma A.13. Let z ∈ H. Define Pt (z) by
Pt (z) = P (Zt − ξt1 , ξt2 − ξt1 ),
0 6 t < τz .
Then Pt (z) is a martingale for the system of commuting SLEs for any choice of the growth
speeds λj (t) > 0.
Proof. Using the expression (A.28) for P , we have
P (Zt −
ξt1 , ξt2
−
ξt1 )
1
=
cα
Z ∞
Zt −ξt1
Re M(U, ξt2 − ξt1 )dU .
Performing the change of variables u = gt−1 (U + ξt1 ), this becomes
1 ∞
Re M(Ut − ξt1 , ξt2 − ξt1 )gt0 (u)du
cα z
Z
α
α
α
α
1 ∞
=
Re (Im Ut )α−2 (Ut − ξt1 )− 2 (Ut − ξt2 )− 2 (Ūt − ξt1 )1− 2 (Ūt − ξt2 )1− 2
cα z
× J(Ut − ξt1 , ξt2 − ξt1 )gt0 (u)du ,
0 6 t < τz ,
Z
Pt (z) =
90
where Ut := gt (u). Now
J(Ut −
ξt1 , ξt2
−
ξt1 )
=
Z Ut −ξ 1
t
Ūt −ξt1
α
α
(V − Ut + ξt1 )α (V − Ūt + ξt1 )α−2 V − 2 (V − ξt2 + ξt1 )− 2 dV,
and the change of variables w = gt−1 (V + ξt1 ) gives
J(Ut −
ξt1 , ξt2
−
ξt1 )
Z u
=
ū
α
α
(Wt − Ut )α (Wt − Ūt )α−2 (Wt − ξt1 )− 2 (Wt − ξt2 )− 2 gt0 (w)dw,
where Wt := gt (W ). Hence
Pt (z) =
1
cα
Z ∞
Z u
Qdwdu ,
Re
0 6 t < τz ,
ū
z
where we use the short-hand notation Q for the integrand which is given by
α
α
α
α
Q := (Im Ut )α−2 (Ut − ξt1 )− 2 (Ut − ξt2 )− 2 (Ūt − ξt1 )1− 2 (Ūt − ξt2 )1− 2
α
α
× (Wt − Ut )α (Wt − Ūt )α−2 (Wt − ξt1 )− 2 (Wt − ξt2 )− 2 gt0 (w)gt0 (u).
Since we have already established smoothness of P , we can apply Itô’s formula to see that
dPt (z) =
1
cα
Z ∞
Z u
z
dQdwdu ,
Re
ū
where
dQ =
∂Q
∂Q
∂Q
∂Q
∂Q
∂Q
dUt +
dWt + 0
dgt0 (u) + 0
dg 0 (w)
dŪt +
dgt0 (u) + 0
∂Ut
∂Wt
∂gt (u)
∂gt (w) t
∂ Ūt
∂gt (u)
∂Q
∂2Q
∂2Q
∂Q
κ
+ 1 dξt1 + 2 dξt2 +
λ1 (t)
+
λ
(t)
.
(A.34)
2
4
(∂ξ 1 )2
(∂ξ 2 )2
∂ξt
∂ξt
Substituting the expressions
dUt =
2
X
λj (t)
j=1
dξt1
j dt,
dŪt =
Ut − ξt
j=1
λ1 (t) + λ2 (t)
dt +
=
ξt1 − ξt2
dgt0 (u) = −gt0 (u)
2
X
j=1
dgt0 (w)
=
−gt0 (w)
2
X
λj (t)
r
(Ut −
Ūt − ξt
κ
λ1 (t)dBt1 ,
2
λj (t)
ξtj )2
dt,
2
X
λj (t)
j=1
(Wt − ξtj )2
j dt,
dξt2
dWt =
2
X
λj (t)
j=1
Wt − ξtj
λ1 (t) + λ2 (t)
dt +
=
ξt2 − ξt1
dgt0 (u) = −gt0 (u)
r
2
X
λj (t)
j=1
(Ūt − ξtj )2
dt,
κ
λ2 (t)dBt2 ,
2
dt,
dt,
into (A.34), a long but straightforward computation shows that the drift term in dQ
vanishes. This shows that Pt (z) is a local martingale and since P (z, ξ) is bounded it is
actually a martingale.
91
θ2
θ2
π
π
4
δ
c
S2
δ
S2
δ
S1
c
δ
c
θ1
0
θ1
Figure 14. The asymptotic sectors S1 and S2 .
B
Estimates for Green’s function
The purpose of this appendix is to prove Lemma 6.2. By applying the method developed
in Section 10, we can determine the asymptotic behavior of h(θ1 , θ2 ) near the boundary
of ∆. This will prove Lemma 6.2.
B.1
The asymptotic sectors
Given δ > 0 and c > 0, we define the open subsets Sj , j = 1, . . . , 5, of ∆ by (see Figures
14, 15, and 16)
√
S1 = {(θ1 , θ2 ) ∈ ∆ | 0 < θ2 − θ1 < c 2, θ1 > δ, θ2 < π − δ},
o
θ1
π
<
−
δ
θ2
4
n
o
θ1
π
∪ (θ1 , θ2 ) ∈ ∆ θ2 > π − c, arctan
<
−
δ
,
π − θ2
4
n
o
π − θ2
π
π − θ2
π
S3 = (θ1 , θ2 ) ∈ ∆ θ2 > π − c, arctan
< − δ, arctan
< −δ ,
1
1
θ
4
π−θ
4
1
n
o
√
θ
π
S4 = (θ1 , θ2 ) ∈ ∆ θ1 + θ2 < c 2, δ < arctan 2 <
θ
4
n
o
√
π − θ2
π
∪ (θ1 , θ2 ) ∈ ∆ θ2 − θ1 > π − c 2, δ < arctan
<
−
δ
θ1
2
2
n
√
π−θ
πo
∪ (θ1 , θ2 ) ∈ ∆ θ1 + θ2 > 2π − c 2, δ < arctan
<
,
π − θ1
4
n
n
S2 = (θ1 , θ2 ) ∈ ∆ θ2 < c, arctan
o
S5 = (θ1 , θ2 ) ∈ ∆ θ1 < c, θ2 − θ1 > δ, θ1 + θ2 < π − δ .
92
θ2
θ2
π
π
c
δ
δ
S3
S4
c
δ
δ
S4
c
δ
δ
S4
c
0
θ1
0
θ1
Figure 15. The asymptotic sectors S3 and S4 .
The asymptotics of h(θ1 , θ2 ) as (θ1 , θ2 ) approaches the boundary of the triangle ∆, can be
described in terms of the five asymptotic sectors {Sj }51 . Indeed, the first four sectors Sj ,
j = 1, . . . , 4, correspond to the different asymptotic regions studied in Sections 10.3-10.6
of Section 10, respectively, while the sector S5 corresponds to the region where w2 → ∞.
Lemma B.1. Let δ > 0. Then there exist constants c > 0 and > 0 such that the
following estimates hold:
(1) dist(w1 , {0, 1, ∞}) > and |w2 − 1| < 1 − for all (θ1 , θ2 ) ∈ S1 ,
(2) |w1 | < 1 − and |w2 | > 1 + for all (θ1 , θ2 ) ∈ S2 ,
(3) |w1 | < 1 − and |w2 | < 1 − for all (θ1 , θ2 ) ∈ S3 ,
(4) |w1 | < 1 − and dist(w2 , {0, ∞}) > for all (θ1 , θ2 ) ∈ S4 ,
(5) dist(w1 , {0, 1, ∞}) > and |w2 | > 1 + for all (θ1 , θ2 ) ∈ S5 .
Proof. The proof follows easily from the definition (9.3) of w1 and w2 .
B.2
Representations for h
The following lemma provides four representations for h which are suitable for determining
the behavior of h for (θ1 , θ2 ) ∈ Sj , j = 1, . . . , 4, respectively. For (θ1 , θ2 ) ∈ S5 , we will use
the representation (9.2) instead.
Lemma B.2. Suppose α > 2 satisfies
h(θ1 , θ2 ) =
3α
2 , 2α
∈
/ Z. Then, for all (θ1 , θ2 ) ∈ ∆,
h
i
iπα
1
2
sinα−1 (θ1 )Im σ(θ2 )(−eiθ )α−1 e− 2 P1 (w1 , w2 ) ,
ĉ
93
(B.1)
θ2
δ
S5
c
δ
θ1
Figure 16. The asymptotic sector S5 .
h
i
1
2
sinα−1 (θ1 )Im σ(θ2 )(−eiθ )α−1 Q1 (w1 , w2 ) + w12α−1 Q2 (w1 , w2 ) ,
ĉ
h
α
1
2
h(θ1 , θ2 ) = sinα−1 (θ1 )Im σ(θ2 )(−eiθ )α−1 R1 (w1 , w2 ) + w22 R2 (w1 , w2 )
ĉ
i
h(θ1 , θ2 ) =
+ w12α−1 Q2 (w1 , w2 ) ,
h(θ1 , θ2 ) =
(B.2)
(B.3)
i
iπα
1
2
sinα−1 (θ1 )Im σ(θ2 )(−eiθ )α−1 (e− 2 T1 (w1 , w2 ) + w12α−1 Q2 (w1 , w2 )) ,
ĉ
(B.4)
h
where w1 , w2 are given by (9.3).
Proof. Equations (B.2) and (B.3) follow from (9.2) together with the expressions (10.22)
and (10.26) for F , respectively. Moreover, suppose we can show that
Im X(θ1 , θ2 ) = 0 in ∆,
(B.5)
where
2
X(θ1 , θ2 ) = σ(θ2 )(−eiθ )α−1 e−
iπα
2
(w2 − 1)1−α P2 (w1 , w2 ),
(B.6)
and the variables w1 and w2 in (B.6) are given by (9.3). Then equations (B.1) and (B.4)
follow from (9.2) together with the expressions (10.17) and (10.31) for F̃ and F = ρF̃ ,
respectively. It therefore only remains to show (B.5).
From the definition (10.16) of P2 we see that for w1 ∈ C \ [0, ∞) and w2 ∈ (0, 1) we
have
P2 (w1 , w2 + i0) =
eiπ(a+d) sin(aπ) −iπ(c+d+1)
e
|1 − w2 |a+b
sin(π(d + c))
94
×
Z (0+,1+,0−,1−) s−1−
A
1 a w1 − 1 b c
s−1+
s (1 − s)d ds,
w2 − 1
w2 − 1
(B.7)
where a, b, c, d are given by (10.4). By definition, the value of P2 (w1 , w2 ) at a general
point (w1 , w2 ) ∈ D1 is determined by analytic continuation of (B.7) within the connected
set D1 ⊂ C2 . The branches of the complex powers in (B.7) are fixed by requiring that the
principal branch is used initially at s = A. This means that whenever the points
A−1−
1
w2 − 1
and A − 1 +
w1 − 1
,
w2 − 1
(B.8)
cross the negative real axis during the analytic continuation, extra factors of e±2πia and
e±2πib , respectively, have to be inserted in (B.7).
In order to evaluate the function X in (B.6), we need the value of P2 at points
(w1 , w2 ) ∈ E, where E denotes the subset of C2 characterized by (9.3), i.e.,
n
2
E = (w1 , w2 ) = 1 − e−2iθ ,
o
sin θ2 −i(θ2 −θ1 ) 1 2
.
(θ
,
θ
)
⊂
∆
e
sin θ1
If w1 and w2 are given by (9.3), then
cot θ2 < cot θ1 ,
1
cot θ2 + i
=
,
w2 − 1
cot θ1 − cot θ2
−
w1 − 1
cot θ2 − i
=
.
w2 − 1
cot θ1 − cot θ2
Hence, we have, for all (w1 , w2 ) ∈ E,
Im A − 1 −
1 < 0,
w2 − 1
Im A − 1 +
w1 − 1 > 0.
w2 − 1
(B.9)
This shows that neither of the points in (B.8) crosses the negative real axis as long as
(w1 , w2 ) remains within E. We can therefore find a formula for P2 valid in E as follows.
Let (w1 , w2 ) be a point in E corresponding to (θ1 , θ2 ) via (9.3). Then
w2 = 1 +
sin(θ2 − θ1 ) −iθ2
e
.
sin θ1
(B.10)
Let 0 < < |w1 − 1| be small and let (w̃1 (t), w̃2 (t)), t ∈ [0, 1], be the path in D1 defined by
w̃1 (t) = w1 for all t, while the path w̃2 (t) starts at 1 − + i0, proceeds clockwise around
2
the small circle of radius centered at 1 until it reaches the point 1 + e−iθ , and then
2
proceeds along the segment [1 + e−iθ , w2 ] until it reaches w2 .
2
1
As w̃2 moves along the arc from 1 − + i0 to 1 + e−iθ , the point A − 1 − w̃2 (t)−1
crosses the negative real axis from the upper into the lower half-plane once (this adds a
1 (t)−1
factor of e2πia to (B.7)), and, provided that Im w1 6 0 (i.e. θ2 > π/2), A − 1 + w̃
w̃2 (t)−1
also crosses the negative real axis from the upper into the lower half-plane once (this adds
w̃1 (t)−1
a factor of e2πib to (B.7)). If Im w1 > 0, then A − 1 + w̃
does not cross the negative
2 (t)−1
1
real axis. By varying θ in (B.10), we see that the part of the path for which w̃2 belongs
2
to the segment [1 + e−iθ , w2 ] lies in E; hence the analytic continuation along this part
adds no more factors to (B.7). We end up with the following formula for P2 in E:
P2 (w1 , w2 ) =
eiπ(a+d) sin(aπ) −iπ(a+b+c+d+1)
e
(w2 − 1)a+b e2πia
sin(π(d + c))
95
×
Z (0+,1+,0−,1−) 1 a w1 − 1 b c
s−1+
s (1 − s)d ds
w2 − 1
w2 − 1
s−1−
A
(
×
e2πib , Im w1 6 0,
1,
Im w1 > 0,
(w1 , w2 ) ∈ E.
Substituting this formula into (B.6) and simplifying, we find
2
X(θ2 , θ2 ) = (−eiθ )α−1 (w2 − 1)α−1 e2πiα
×
Z (0+,1+,0−,1−) s−1−
A
α
1 α−1 w1 − 1 α−1 − α
s 2 (1 − s)− 2 ds,
s−1+
w2 − 1
w2 − 1
where w1 , w2 are given by (9.3). But
2
(−eiθ )α−1 (w2 − 1)α−1 = (sinα−1 θ2 )(cot θ1 − cot θ2 )α−1 e−πi(α−1) ,
and, by (B.9),
s−1−
1 α−1 w1 − 1 α−1 1 + ((1 − s) cot θ1 + s cot θ2 )2 α−1
=
.
s−1+
w2 − 1
w2 − 1
(cot θ1 − cot θ2 )2
Hence
X(θ2 , θ2 ) = − sinα−1 (θ2 )(cot θ2 − cot θ1 )α−1 eπiα
Z (0+,1+,0−,1−) α
1 + ((1 − s) cot θ1 + s cot θ2 )2 α−1 − α
×
s 2 (1 − s)− 2 ds.
1
2 2
(cot θ − cot θ )
A
(B.11)
If g(s) is an analytic function, the identity (9.16) implies
Z (0+,1+,0−,1−)
g(s)sc (1 − s)d ds =
Z (0−,1−,0+,1+)
g(s̄)sc (1 − s)d ds
A
A
−1 + e−2πic − e−2πi(c+d) + e−2πid
=
−1 + e2πic − e2πi(c+d) + e2πid
= e−2πi(c+d)
Z (0+,1+,0−,1−)
Z (0+,1+,0−,1−)
g(s̄)sc (1 − s)d ds
A
g(s̄)sc (1 − s)d ds.
A
Using this identity to compute the imaginary part of (B.11) we arrive at
Im X(θ1 , θ2 ) = − sinα−1 (θ2 )(cot θ2 − cot θ1 )α−1
Z (0+,1+,0−,1−)
α
A
×
1 + ((1 − s) cot θ 1 + s cot θ 2 )2 α−1
(cot θ1
−
cot θ2 )2
α
96
α
s− 2 (1 − s)− 2 ds = 0.
This proves (B.5) and completes the proof of the lemma.
α
eπiα − e−πiα e−2iπ(− 2 − 2 )
B.3
Proof of Lemma 6.2
We are now in a position to prove Lemma 6.2. Indeed, since h clearly is smooth in the
interior of ∆ and the parameter δ > 0 which defines the sectors Sj is arbitrary, Lemma
6.2 is a direct consequence of the following result.
Lemma B.3. Let α > 2. Then the function h(θ1 , θ2 ) defined in (6.11) satisfies the
following estimates:
|h(θ1 , θ2 )| 6 C sinα−1 θ1 ,
|h(θ1 , θ2 )
− hf
α−1
sin
θ1
(θ2 )|
(θ1 , θ2 ) ∈ ∪5j=1 Sj ,
|θ2
θ1 |
−
,
sin θ1
|h(θ1 , θ2 ) − sinα−1 θ1 |
sin θ2
,
6
C
sin θ1
sinα−1 θ1
6C
(B.12)
(θ1 , θ2 ) ∈ S1 ,
(B.13)
(θ1 , θ2 ) ∈ S3 ,
(B.14)
where hf (θ) is defined in (8.8).
Proof. Let us first assume that α > 2 satisfies 3α
/ Z. Equation (B.1), Lemma B.1
2 , 2α ∈
(1), and Lemma 10.5 (a) show that (B.12) holds in S1 . Also, by Lemma 10.5 (a), the
following estimate is valid in S1 :
|h(θ1 , θ2 ) − h(θ2 , θ2 )|
|θ2 − θ1 |
sin(θ2 − θ1 )
6
C
,
6
C|w
−
1|
=
C
2
sin θ1
sin θ1
sinα−1 θ1
(B.15)
where
h
i
iπα
1
2
sinα−1 (θ2 )Im σ(θ2 )(−eiθ )α−1 e− 2 P1 (w1 , 1) .
ĉ
For a, b, c, d given by (10.4), we have (cf. (8.13))
h(θ2 , θ2 ) =
P1 (w1 , 1) =
e−iπc sin(dπ)
sin(π(d + c))
Z (0+,1+,0−,1−)
iπα
= 2iπ(−1 + e
v a (v − w1 )b (1 − v)c+d dv
A
α−1
)(−w1 )
1
1 − α, α; 1;
.
w1
2 F1
It follows that h(θ1 , θ1 ) = hf (θ1 ) where hf is given by (8.8). Equation (B.13) then follows
from (B.15).
1 2
2
Using the fact that dist( w
w1 , {0, 1}) > for all (θ , θ ) ∈ ∆, Lemma B.1 (2) and Lemma
10.5 (b) and (c) imply
α
|Q1 + w12α−1 Q2 | 6 C|w2 |− 2 + C|w1 |
3α
−1
2
6 C,
(θ1 , θ2 ) ∈ S2 .
Hence equation (B.2) shows that (B.12) holds in S2 .
Similarly, Lemma B.1 (3), and Lemma 10.5 (c), (d), and (e) show that
α
|R1 + w22 R2 + w12α−1 Q2 | 6 C + C|w2 |
3α
−1
2
+ C|w1 |
3α
−1
2
6 C,
(θ1 , θ2 ) ∈ S3 .
Hence equation (B.3) implies that (B.12) holds in S3 . Also, by Lemma 10.5 (d), since
α > 2, the following estimate is valid in S3 :
3α
3α
|h(θ1 , θ2 ) − h(θ1 , π)|
6 C(|w1 | + |w2 |) + C|w2 | 2 −1 + C|w1 | 2 −1 6 C(|w1 | + |w2 |)
α−1
1
sin
θ
97
sin θ 2 sin θ 2 6
C
,
1
1
6 C|π − θ2 | + C sin θ
sin θ
where
sinα−1 θ1
Im e−iπα R1 (0, 0) .
ĉ
For a, b, c, d given by (10.4), we have
h(θ1 , π) =
(0+,1+,0−,1−)
e2πia − 1
v a+b+c (1 − v)d dv
R1 (0, 0) = 2πi(a+b+c)
e
−1 A
(e2iπa − 1)(e2iπd − 1)Γ(d + 1)Γ(a + b + c + 1)
= −
Γ(a + b + c + d + 2)
2iπα
−iπα
(e
− 1)(e
− 1)Γ(1 − α2 )Γ( 3α
2 − 1)
= −
.
Γ(α)
Z
Taking the definition (2.9) of ĉ into account, it follows that h(θ1 , π) = sinα−1 θ1 . This
proves (B.14).
Lemma B.1 (4), and Lemma 10.5 (c) and (f ) show that
|e−
iπα
2
α
T1 + w12α−1 Q2 | 6 C + C|w1 |− 2 6 C,
(θ1 , θ2 ) ∈ S4 .
Hence equation (B.4) implies that (B.12) holds in S4 .
Lemma B.1 (5), and Lemma 10.5 (g) show that
|F | 6 C|w2 |
3α
−1
2
6 C,
(θ1 , θ2 ) ∈ S5 .
Hence equation (9.2) shows that (B.12) holds in S5 . This completes the proof of the
lemma in the case when 3α
2 and 2α are not integers.
3α
Assume finally that 2 and/or 2α is an integer. Then some of the functions in Lemma
10.5 degenerate, so a slightly different argument is required. We do not give complete
details, but outline the relevant steps.
Suppose first that α ∈
/ Z but 3α
2 or 2α is an integer. Then the limit w2 → 1 can
still be treated as in Section 10, because c + d = −α is not an integer. However, the
limits involving w1 → 0 or w2 → 0 cannot be treated in the same way in general, because
a + b = 2α − 2 and/or a + b + c = 3α
2 − 2 is an integer. However, since α > 2, we have
a+b > 0 and a+b+c > 0. Hence the integral (9.1) defining F is nonsingular at v = 0 (also
in the limit as w1 and w2 approach zero). Hence, we can derive the leading behavior of F
in these regimes using the following alternative approach: First, we collapse the two loops
of the Pochhammer contour enclosing the origin down to the interval [0, A] (cf. equation
(9.12)). Then we find the leading-order asymptotics by Taylor expanding the integrand
as w1 and/or w2 approaches zero.
Assume finally that α = n > 2 is an integer. This case is considered in Section 9,
where an expression for h(θ1 , θ2 ; n) is derived by taking the limit of the defining equation
(9.1) for F as α → n. In order to prove (B.12)-(B.14) in this case, we compute the limits
as α → n of each of the four equations in Lemma B.2. This gives four analogous equations
valid for α = n. As above, it follows from these equations that h satisfies (B.12)-(B.14).
The crucial point is that the singular contribution from P2 vanishes as a consequence of
(B.5).
98
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