arXiv:1211.2451v1 [math-ph] 11 Nov 2012
The Coefficient Problem and Multifractality of
Whole-Plane SLE & LLE
Bertrand Duplantier (a,b)∗, Nguyen Thi Phuong Chi (c)†,
Nguyen Thi Thuy Nga‡(c) , Michel Zinsmeister (c,d)§
(a)
Institut de Physique Théorique, CEA/Saclay
F-91191 Gif-sur-Yvette Cedex, France
(b)
The Mathematical Sciences Research Institute
17 Gauss Way, Berkeley, CA 94720-5070, USA
(c)
MAPMO
Université d’Orléans
Bâtiment de mathématiques, rue de Chartres
B.P. 6759 - F-45067 Orléans Cedex 2, France
(d)
Laboratoire de physique théorique de la matière condensée
UMR CNRS 7600, Tour 12-13/13-23, Boı̂te 121
4, Place Jussieu
75252 Paris Cedex 05, France
November 13, 2012
Abstract
∗
e-mail:
CSD5.
†
e-mail:
‡
e-mail:
grant APR
§
e-mail:
[email protected] – Partially supported by grant [email protected]
[email protected] – Partially supported by the “Région Centre” through the
TRUC (Transport, Réseaux, Croissance, Urbanisme.)
[email protected]
1
Karl Löwner (later known as Charles Loewner) introduced his famous
differential equation in 1923 in order to solve the Bieberbach conjecture for
series expansion coefficients of univalent analytic functions at level n = 3.
His method was revived in 1999 by Oded Schramm when he introduced the
Stochastic Loewner Evolution (SLE), a conformally invariant process which
made it possible to prove many predictions from conformal field theory for
critical planar models in statistical mechanics. The aim of this paper is to
revisit the Bieberbach conjecture in the framework of SLE processes and,
more generally, Lévy processes. The study of their unbounded whole-plane
versions leads to a discrete series of exact results for the expectations of coefficients and their variances, and, more generally, for the derivative moments
of some prescribed order p. These results are generalized to the “oddified”
or m-fold conformal maps of whole-plane SLEs or Lévy-Loewner Evolutions
(LLEs). We also study the (average) integral means multifractal spectra of
these unbounded whole-plane SLE curves. We prove the existence of a phase
transition at a moment order p = p∗ (κ) > 0, at which one goes from the bulk
SLEκ average integral means spectrum, as predicted by one of us [17] and
established by Beliaev and Smirnov [4], and valid for p ≤ p∗ (κ), to a new
integral means spectrum for p ≥ p∗ (κ), as partially conjectured in Ref. [49].
The latter spectrum is furthermore shown to be intimately related, via the
associated packing spectrum, to the radial SLE derivative exponents obtained
by Lawler, Schramm and Werner [42], and to the local SLE tip multifractal
exponents obtained from quantum gravity in Ref. [19]. This is generalized to
the integral means spectrum of the m-fold transform of the unbounded wholeplane SLE map. A succinct, preliminary, version of this study first appeared
in Ref. [23].
1
1.1
Introduction
The coefficient problem and Schramm-Loewner evolution
P
Let f (z) = n≥0 an z n be a holomorphic function in the unit disc D. We further
assume that the function f is injective: what then can be said about the coefficients
an ? An trivial observation is that a1 6= 0 and Bieberbach [7] proved in 1916 that
|a2 | ≤ 2|a1 |.
In the same paper he famously conjectured that
∀n ≥ 2, |an | ≤ n|a1 |,
guided by the intuition that the function (afterwards called the Koebe function)
X
z
K(z) := −
n(−z)n =
,
(1)
(1 + z)2
n≥1
2
which is a holomorphic bijection between D and C\[1/4, +∞), should be extremal.
This conjecture was finally proven in 1984 by De Branges [12]: its proof was made
possible by the addition of a new idea (an inequality of Askey and Gasper) to a series
of methods and results developed in almost a century of effort. It is largely accepted
that the earliest important contribution to the proof of Bieberbach’s conjecture is
the proof [50] by Loewner in 1923 that |a3 | ≤ 3|a1 |. De Branges’ proof in 1985 [12]
indeed used Loewner’s idea in a crucial way, as did many contributors to the proof
around that time. In an Appendix to this article, we recall the proof by Bieberbach
for the case n = 2, and that by Loewner for n = 3. It ends with a brief account of
post-Loewner steps towards the proof of Bieberbach’s conjecture.
Loewner’s ideas go far beyond Bieberbach’s conjecture: Oded Schramm [63]
revived Loewner’s method in 1999, introducing randomness into it, as driven by
standard Brownian motion. This field, now called the theory of SLE processes
(initially for Stochastic Loewner, now for Schramm-Loewner, Evolution), provides
a unified and rigorous approach to the geometry of conformally invariant processes
and critical curves in two-dimensional statistical mechanics. It led to the two Fields
medals of W. Werner (for the application of SLE to planar Brownian paths) and of
S. Smirnov (for application of SLE to critical percolation and Ising models).
The aim of the present paper is to revisit Bieberbach’s conjecture in the framework of SLE theory, that is to study the coefficients of univalent functions coming
from the conformal maps associated with this process. We also extend our study
to the so-called Lévy-Loewner Evolution (LLE), where the Brownian source term in
Loewner’s equation is generalized to a Lévy process. (See, e.g.,[62, 55].)
There exist several variants of SLEκ , known, in a terminology due to Schramm,
as chordal, radial, or whole-plane. The one we adopt in this work is a variant of
the whole-plane one, corresponding to the original setting introduced by Loewner.
As in the radial case, the whole-plane Loewner process is determined by a function
λ : [0, +∞) → ∂D := {z : |z| = 1}, called the driving function, obtained as follows.
Define γ : [0, ∞) → C to be a Jordan arc joining γ(0) to ∞, and not containing the
origin 0 (see Fig. 1). Define then for each t > 0, the slit domain Ωt = C\γ([t, ∞)).
It is a simply connected domain containing 0 and we can thus consider the Riemann
mapping ft : D → Ωt , ft (0) = 0, ft0 (0) > 0. By the Caratheodory convergence
theorem, ft converges as t → 0 to f := f0 , the Riemann mapping of Ω0 . We may
assume without loss of generality that f 0 (0) = 1 and, by changing the time t if
necessary, choose the normalization ft0 (0) = et .
The key idea of Loewner is to use the fact that the sequence of domains Ωt is
t
t
/z ∂f
> 0 or, equivalently, that
increasing, which translates into the fact that < ∂f
∂t
∂z
this quantity is the Poisson integral of a positive measure on the unit circle, actually
a probability measure because of the above normalization. Now the fact that the
domains Ωt are slit domains implies that for every t this probability measure must
be a Dirac mass at point λ(t) = ft−1 (γ(t)). It is worthwhile to notice that λ is a
continuous function. One says that the Loewner chain (ft ) associated with (Ωt ) is
3
Figure 1: Loewner map z 7→ ft (z) from D to the slit domain Ωt = C\γ([t, ∞)) (here
slit by a single curve γ([t, ∞)) for κ ≤ 4). One has ft (0) = 0, ∀t ≥ 0. At t = 0, the
driving function λ(0) = 1, so that the image of z = 1 is at the tip γ(0) = f0 (1) of
the curve.
driven by the function λ(t), in the sense that ft satisfies the Loewner differential
equation
∂ft λ(t) + z
∂ft
=z
, z ∈ D.
(2)
∂t
∂z λ(t) − z
It is remarkable that that the Loewner method can be reversed: given a function
λ which is càdlàg, i.e., right continuous with left limits at every point of R+ with
values in the unit circle, then the Loewner equation (2) has a solution ft (z) which is
the Riemann mapping of a domain Ωt and the corresponding family is increasing in t.
As is well-known, Schramm’s fundamental insight was to consider as a particular
driving function
√
λ(t) = ei κBt ,
(3)
where κ ∈ [0, ∞), and Bt is standard, one-dimensional, Brownian motion, characterized by the three fundamental properties:
4
(a) Stationarity: if 0 ≤ s ≤ t, then Bt − Bs has the same law as Bt−s ;
(b) Markov property: if 0 ≤ s ≤ t, then Bt − Bs is independent of Bs ;
(c) Gaussianity: Bt has a normal distribution with mean 0 and variance t.
A Lévy process is the generalization that is assumed to satisfy only the first two
of these properties, the essential difference with Brownian motion being that jumps
are then allowed. The characteristic function of a Lévy process Lt has the form
E(eiξLt ) = e−tη(ξ)
(4)
where η (called the Lévy symbol) is a continuous complex function of ξ ∈ R,
satisfying (in addition to necessary Bochner type conditions [2]) η(0) = 0, and
η(−ξ) = η(ξ). SLEκ corresponds to a Gaussian characteristic function, and is a
Lévy process with symbol
η(ξ) = κξ 2 /2.
(5)
More generally, the function
η(ξ) = κ|ξ|α /2, α ∈ (0, 2]
(6)
is the Lévy symbol of the so-called α−stable process. The normalization here is
chosen so that this process is SLEκ for α = 2. Another Lévy symbol of interest
is given (up to constant factor) by η(ξ) = 1 − (sin πξ)/πξ, and corresponds to a
certain compound Poisson process which serves as a model for a dendritic growth
process; this aspect will be developed in a forthcoming paper (see also [35]). In the
examples above η is a real, therefore even function, a property which we will assume
throughout, except in the beginning of Section 2. As we shall see, all the quantities
that we will consider depend only on the values of the Lévy symbol at integer
arguments; for this reason we shall use the “sequence” notation: ηk := η(k), k ∈ Z.
The associated conformal maps, obeying (2), are denoted by ft , and in this
work, we study their coefficients an (t), which P
are random variables,
defined by the
normalized series expansion: ft (z) = et z + n≥2 an (t)z n . Section 2 starts with
the computation, in terms of the Lévy symbols ηk , k ∈ Z, of E(an ) for all n, and of
E(|an |2 ) for small n, for a general Lévy-Loewner evolution process ft . Note that a
similar idea already appeared in Ref. [38], where A. Kemppainen studied in detail
the coefficients associated with the Schramm-Loewner evolution, using a stationarity
property of SLE [40]. However, the focus there was on expectations of the moments
of those coefficients, rather than on the moments of their moduli.
p
2
2 represented
We also consider the associated odd process: ht (z) :=
z
P ft (z )/z ),
−t/2
by the normalized series expansion: e
ht (z) = z + n≥1 b2n+1 (t)z 2n+1 , thus investigating the famous Fekete-Szegö phenomenon in the stochastic realm. We find
the following results:
5
Theorem 1.1. Let ft be the whole-plane evolution generated by the Lévy process
(Lt ) with Lévy symbol η. We write
e−t ft (z) = z +
∞
X
an (t)z n ; e−t/2 ht (z) = z +
n=2
X
b2n+1 (t)z 2n+1 .
n≥1
Then the conjugate whole-plane Lévy-Loewner evolution e−iLt ft eiLt z has the same
(law)
law as f0 (z), i.e., ei(n−1)Lt an (t) = an (0). Similarly,
the conjugate oddified whole
plane Lévy-Loewner evolution e−(i/2)Lt ht e(i/2)Lt z has the same law as h0 (z), i.e.,
(law)
einLt bn (t) = bn (0).
Setting an := an (0) and b2n+1 := b2n+1 (0), we have
E(an ) =
E(b2n+1 ) =
n−2
Y
ηk − k − 2
, n ≥ 2,
η
k+1 + k + 1
k=0
n−1
Y
ηk − k − 1
, n ≥ 1.
ηk+1 + k + 1
k=0
Corollary 1.1. In the setting of Theorem 1.1,
(i) if η1 = 3, E(f00 (z)) = 1 − z;
(ii) if η1 = 1 and η2 = 4, E(f00 (z)) = (1 − z)2 ;
(iii) if η1 = 2, E(ϕ00 (z)) = 1 − z 2 .
Computations of expectations E(|an |2 ) are already quite involved at level n = 4,
and we have used computer assistance in symbolic calculus with matlab for higher
coefficients. These computer experiments, briefly explained in Section 2.2, lead to
the following statements, explicitly checked up to n = 8:
Theorem 1.2. In the same setting as in Theorem 1.1,
(i) if η1 = 3, E(|an |2 ) = 1, n ≥ 1;
(ii) if η1 = 1∗ , E(|an |2 ) = n, n ≥ 1;
(iii) if η1 = 2, E(|b2n+1 |2 ) = 1/(2n + 1), n ≥ 1.
Remarks 1.1. For SLEκ , Eq. (5) gives η1 = κ/2, thus case (i) includes SLEκ=6 .
The special notation for case (ii), η1 = 1∗ , has a dual meaning: this case is directly
verified for η1 = 1 and all n ≤ 8, and proven for all n under the further assumption
that η2 = 4. Since Eq. (5) further gives η2 = 2κ, SLE2 is included in that rigorous
case. Case (iii) includes SLE4 .
6
Section 3 is devoted to proofs and begins with the computation of E(ft (z)) and
E(ht (z)). We show in particular that these expectations take a simple, polynomial
form for the two cases above, η1 = 3 and η1 = 1, η2 = 4, and more generally, when
there exists a k ∈ N, such that ηk = 2 + k. In the odd case, these special values
are ηk = 1 + k. This also yields the derivative expectations E[ft0 (z)]. These results
are used in the remainder of the section, devoted to proving Theorem (1.2) and
obtaining other identities.
After our earlier draft was posted [23], the SLE cases (i) and (ii) of Theorem
1.2 were obtained in a draft communicated to us by our former collaborator, O.
Yermolayeva, together with I. Loutsenko, later transformed into Ref. [49]. Their
derivation was based on the use of a differential equation obeyed by the expected
moments of |ft0 (z)|, and obtained by a heuristic method originally due to M. Hastings
[4]. It allowed them to derive, by series expanding the second moment, a double
recursion acting on expectations E(an an0 ), which can be solved for κ = 6 or 2,
apparently with the help of maple.
The differential equation actually appeared earlier, together with its rigorous
proof, in a paper by D. Beliaev and S. Smirnov [4] (based on Beliaev’s PhD dissertation [3]), for another variant of the whole-plane SLE, and with an explicit
extension to the LLE case. The latter extension allows us to extend the proof of
cases (i) and (ii) to Lévy-Loewner evolutions.
Starting from the Beliaev-Smirnov (BS) equation, and using the further information gained from the computation of expectations of derivatives (and moments thereof), we provide here a simple analytical way to obtain a series of explicit solutions of that equation, which were first observed in Ref.
0[49].p/2In
the
case of SLEκ , exact expressions are obtained for the moments E (ft (z))
and
0
0
p/2
p
p/2
0
, for a special set of values of the parameters p
E |ft (z)| = E (ft (z)) (ft (z̄))
and κ, that includes p = 2 and cases (i) and (ii). This results in a direct proof of the
main statements above and of other identities. It circumvents the use of a double
recursion, and does not require computer assistance. We also derive modified BS
equations for the “oddified” version of the SLE or LLE processes, or for their m-fold
transforms, the exact solutions of which yield in particular a proof of case (iii).
We would like to stress that it is only for the “inner” variant of whole-plane
SLE or LLE that we have introduced in Ref. [23] and study here, that such explicit,
closed-form properties may exist.
1.2
Integral means spectra of whole-plane SLE
These results are used in Section 4, to study the multifractal integral means spectrum of our whole-plane processes. Plancherel’s theorem yields the easy corollary of
Theorem 1.2:
7
Corollary 1.2. For a Lévy-Loewner evolution with η1 = 0, 1∗ , 3 (thus including SLE
for κ = 0, 2, 6), and for an oddified LLE with η1 = 2 (thus SLE for κ = 4), one has,
respectively:
Z 2π
1 + 11r2 + 11r4 + r6 1 + 4r2 + r4 1 + r2
1 + r4
1
0
iθ 2
|f (re )| dθ =
;
;
;
.
E
2π 0
(1 − r2 )5
(1 − r2 )4 (1 − r2 )3 (1 − r4 )2
The first case is obtained directly from the Koebe function (1), which coincides with
the whole-plane SLE map for κ = 0. We can rephrase these results in terms of the
following:
Definition 1.1. The integral means spectrum of a conformal mapping f is the
function defined on R by
R
log( ∂D |f 0 (rz)|p |dz|)
.
(7)
β̂(p) := limr→1
1
log( 1−r
)
In the stochastic setting, we define the average integral means spectrum
Definition 1.2.
R
log( ∂D E |f 0 (rz)|p |dz|)
β(p) := limr→1
.
1
log( 1−r
)
(8)
The preceding results show that, in the expectation sense of definition (8), these
exponents can be read off as β(2) = 5, 4, 3 for whole-plane LLE with η1 = 0, 1∗ , 3
(thus whole-plane SLE with κ = 0, 2, 6) respectively. For the oddified LLE with
η1 = 2 (thus the oddified whole-plane SLE4 ), β2 (2) = 2.
Remark 1.1. Define the functions
p
1 2
4 + κ − (4 + κ) − 8κp ,
γ0 (p, κ) :=
2κ
κ 2
4+κ
β0 (p, κ) :=
γ0 = −p +
γ0
2
2
p
4+κ
2
= −p +
4 + κ − (4 + κ) − 8κp ,
4κ
(4 + κ)2
β̂0 (p, κ) := p −
.
16κ
(9)
(10)
(11)
They yield the average integral means spectrum β̄0 (p, κ) of the bulk of the outer
whole-plane version of SLEκ , as given by Eqs. (11) (12) and (14) in Beliaev and
Smirnov [4]:
β̄0 (p, κ) = β0 (p, κ), 0 ≤ p ≤ p∗0 (κ),
= β̂0 (p, κ), p ≥ p∗0 (κ),
3(4 + κ)2
p∗0 (κ) :=
.
32κ
8
(12)
(13)
(14)
Motivated by the observations above, we determine, from the singularity analysis
near the unit circle of the BS equation, or of its oddified or m-fold version, the
integral means spectrum of our inner version of the whole-plane SLEκ . It is given
in the following (non-rigorous) statement:
Statement 1.1. In the unbounded case of the inner whole-plane SLEκ process,
ft=0 (z), z ∈ D, as defined by the Schramm-Loewner equation (2), and of its m
1/m
(m)
fold transforms, h0 (z) := z f0 (z m )/z m
, m ≥ 1, the respective average integral
means spectra β(p, κ) and βm (p, κ) all exhibit a phase transition and are given, for
p ≥ 0, and for 1 ≤ m ≤ 3, by
1 1p
1 + 2κp ,
(15)
β(p, κ) := β1 (p, κ) = max β0 (p, κ), 3p − −
2 2
1 1p
1 + κp ,
(16)
β2 (p, κ) = max β0 (p, κ), 2p − −
2 2
(
)
r
2
1 1
2κp
βm (p, κ) = max β0 (p, κ), 1 +
p− −
. (17)
1+
m
2 2
m
The first spectrum β1 has its transition point, where the second term supersedes the
first one, at
p
1 p∗ (κ) = p∗1 (κ) :=
(18)
(4 + κ)2 − 4 − 2 4 + 2(4 + κ)2 ,
16κ
while the second one β2 has its transition point at
p
1 p∗2 (κ) :=
3(4 + κ)2 − 24 − 4 3(4 + κ)2 + 36
36κ
p
1 p
3(4 + κ)2 + 36 − 8
3(4 + κ)2 + 36 + 4 ,
=
36κ
(19)
and in general:
p
m
2
2 + 4m2
(m
+
1)(4
+
κ)
−
8m
−
4
(m
+
1)(4
+
κ)
8κ(m + 1)2
p
m
2 + 4m2 − 2m − 4
=
(m
+
1)(4
+
κ)
8κ(m + 1)2
p
2
2
×
(m + 1)(4 + κ) + 4m + 2m .
(20)
p∗m (κ) :=
For 1 ≤ m ≤ 3, one has ∀κ ≥ 0, p∗m (κ) ≤ p∗0 (κ) so that β0 = β̄0 in (17).
For m ≥ 4, the average integral means spectrum of the unbounded inner wholeplane SLEκ is given by
)
(
r
1 1
2κp
2
βm (p, κ) = max β̄0 (p, κ), 1 +
p− −
1+
,
(21)
m
2 2
m
9
with β̄0 defined as in (12)-(11). Then
p∗m (κ) S p∗0 (κ),
κ S κm , κm := 4
m+3
,
m−3
m ≥ 4,
(22)
such that for κ ≤ κm ,
βm (p, κ) = β0 (p, κ), 0 ≤ p ≤ p∗m (κ),
r
2
1 1
2κp
=
1+
p− −
1+
, p∗m (κ) ≤ p,
m
2 2
m
(23)
(24)
whereas for κ ≥ κm ,
βm (p, κ) = β0 (p, κ), 0 ≤ p ≤ p∗0 (κ),
(κ),
= β̂0 (p, κ), p∗0 (κ) ≤ p ≤ p∗∗
r m
2
1 1
2κp
=
1+
, p∗∗
p− −
1+
m (κ) ≤ p,
m
2 2
m
(25)
(26)
(27)
where p∗∗
m (κ) is the second critical point
p∗∗
m (κ) := m
κ2 − 16
,
32κ
(28)
where the last spectrum (27) supersedes the linear spectrum (13), (26).
For p ≤ 0, the average integral means spectrum, common to all m-fold versions of
the inner or outer whole-plane SLE, is given, as in Eq. (14) of [4], for −1 − 3κ/8 <
p ≤ 0 by the bulk spectrum β0 (p, κ) (10), and for p ≤ −1 − 3κ/8 by the so-called
tip-spectrum [4, 33, 36]:
κ
β̄(p, κ) = βtip (p, κ) := β0 (p, κ) − 2γ0 (p, κ) − 1 = −p − 1 + γ0 (p, κ)
(29)
2
p
1
3κ
= −p +
4 + κ − (4 + κ)2 − 8κp , p ≤ −1 − .(30)
4
8
For the second order moment case p = 2, and for the special cases m = 1,
κ = 0, 2, 6 or m = 2, κ = 4, the expressions (15) and (16) above agree with the
results stated in Corollary 1.2. The rightmost expression in (15) first appeared as a
conjecture in Ref. [49].
As mentioned above, there exists a special point [4]
(6 + κ)(2 + κ)
,
(31)
8κ
where an exact expression can be found for E |f0 (z)|p (Theorem 3.4); more generally there exists a series of special points
p = p(κ) = p1 (κ) :=
p = pm (κ) :=
m(2m + 4 + κ)(2 + κ)
, m ≥ 1,
2(m + 1)2 κ
10
(32)
(m)
where the p-th moment of the m-fold transform, E |h0 (z)|p , is found in an exact
form (Theorems 3.6 and 3.8). Note that p∗m (κ) ≤ pm (κ), ∀κ ≥ 0 and p∗∗
m (κ) ≤
pm (κ), ∀κ ≥ κm .
In this setting, we rigorously prove the following
Theorem 1.3. The average integral means spectrum β(p, κ) := β1 (p, κ) of the unbounded inner whole-plane SLEκ (2) (3) has a phase transition at p∗ (κ) (18) and a
special point at p(κ) (31), such that
β(p, κ) = β0 (p, κ), 0 ≤ p ≤ p∗ (κ);
1 1p
β(p, κ) ≥ 3p − −
1 + 2κp > β0 (p, κ), p∗ (κ) < p < p(κ);
2 2
1 1p
(6 + κ)2
β(p, κ) = 3p − −
1 + 2κp =
, p = p(κ);
2 2
8κ
1 1p
1 + 2κp, p(κ) < p.
β(p, κ) ≤ 3p − −
2 2
Theorem 1.4. Similarly, the average integral means spectrum of the m-fold transform of the unbounded whole-plane SLE map has a phase transition at p∗m (κ) (20)
and a special point at pm (κ) (32), such that for 1 ≤ m ≤ 3, ∀κ, or m ≥ 4, κ ≤ κm =
4(m + 3)/(m − 3),
βm (p, κ) = β0 (p, κ), 0 ≤ p ≤ p∗m (κ);
r
2
1 1
βm (p, κ) ≥
1+
1+
p− −
m
2 2
r
2
1 1
1+
βm (p, κ) =
1+
p− −
m
2 2
r
2
1 1
βm (p, κ) ≤
1+
p− −
1+
m
2 2
2κp
> β0 (p, κ), p∗m (κ) < p < pm (κ);
m
2κp
(2m + 4 + κ)2
=
, p = pm (κ);
m
2(m + 1)2 κ
2κp
, pm (κ) < p.
m
For m ≥ 4 and κ ≥ κm , the average integral means spectrum of the m-fold transform
of the unbounded whole-plane SLEκ map has a phase transition at p∗∗
m (κ) (28)
βm (p, κ) = β̄0 (p, κ), 0 ≤ p ≤ p∗∗
m (κ);
r
2
1 1
βm (p, κ) ≥
1+
p− −
1+
m
2 2
r
2
1 1
βm (p, κ) =
1+
p− −
1+
m
2 2
r
2
1 1
βm (p, κ) ≤
1+
p− −
1+
m
2 2
11
2κp
> β̄0 (p, κ), p∗∗
m (κ) < p < pm (κ);
m
2κp
(2m + 4 + κ)2
=
, p = pm (κ);
m
2(m + 1)2 κ
2κp
, pm (κ) < p.
m
Remark 1.2. The phase transition point p∗m (κ) (18) is lower, for 1 ≤ m ≤ 3, ∀κ,
or for 4 ≤ m, κ ≤ κm , than the special value p∗0 (κ) = 3(4 + κ)2 /32κ after which the
BS spectrum becomes linear in p [4]. The phase transition specific to the unbounded
whole-plane SLEκ then supersedes the usual phase transition towards a linear behavior. For m ≥ 4, κ ≥ κm , the situation is reversed, and the linear transition at p∗0 (κ)
happens before the one specific to the unboundedness of the inner whole-plane SLEκ ,
which thus takes place at the higher value p∗∗
m (κ) (28).
Remark 1.3. As mentioned above, for p ≤ 0, all the average spectra β(p, κ) :=
β1 (p, κ), βm (p, κ), m ≥ 2, coı̈ncide with the one derived by Beliaev and Smirnov [4],
which equals β0 (p, κ) down to the phase transition for p ≤ −1 − 3κ/8 to the tip
spectrum (29)-(30), as predicted in the multifractal formalism in [33] and proven in
an almost sure sense in [36].
In the κ → 0 limit, one has limκ→0 β0 (p, κ) = 0, and the spectra:
β(t, κ = 0) = max{0, 3p − 1},
β2 (t, κ = 0) = max{0, 2p − 1},
βm (t, κ = 0) = max{0, (1 + 2/m)p − 1},
(33)
(34)
(35)
coı̈ncide with those directly derived (for p ≥ 0) for the Koebe function (1) and its
m-fold transforms.
Note also that these integral means spectra at p = 2 give the asymptotic behaviors of the coefficient second√moments: E|an |2 nβ(2)−3 and E|b2n+1 |2√ nβ2 (2)−3 for
n → ∞, with β(2) = (11 − 1 + 4κ)/2 [for κ ≤ 30] and β2 (2) = (7 − 1 + 2κ)/2)/2
[for κ ≤ 24].
Another interesting random variable is the area of the image of the unit disk
Z Z
∞
X
0
2
n|an |2 .
|f (z)| dxdy = π
D
n=1
The expectation of this quantity thus converges for β(2) < 1, i.e., only for κ > 20,
even though the SLE trace is no longer a simple curve as soon as κ > 4. Similarly,
for the odd case, convergence of the area is obtained for β2 (2) < 1, hence for κ > 12.
Derivative exponents
In the so-called multifractal formalism [30, 32, 34, 53], the integral means spectrum
(7), or its in expectation version β(p) (8), are related by various (Legendre) transforms to other multifractal spectra, such as the so-called packing spectrum, the
moment spectrum, often written τ (q) or τ (n), the generalized dimension spectrum
D(n) := τ (n)/(n−1), and the celebrated multifractal spectrum f (α) (see, e.g., Refs.
[4, 19, 32, 52]). Of particular interest here is the packing spectrum [52], defined as
s(p) := β(p) − p + 1.
12
(36)
For our unbounded whole-plane SLEκ , we have for p ≥ p∗ (κ) (15), (18):
1 1p
−
1 + 2κp,
2 2
s(p, κ) = β(p, κ) − p + 1
1 1p
1 + 2κp.
= 2p + −
2 2
β(p, κ) = 3p −
(37)
(38)
(39)
It is then particularly interesting to consider the inverse function of s(p, κ), p(s, κ) =
s−1 (s, κ). It has two branches,
p
1 s
κ − 4 ± (4 − κ)2 + 16κs ,
p± (s, κ) := +
2 16
which are both defined for s ≥ smin (κ) := −(4 − κ)2 /16κ, where they share the
common value pmin (κ) := p± (smin (κ), κ) = (κ−4)(κ+4)/32κ. One has p− (s) ≤ pmin ,
whereas p+ (s) ≥ pmin . Since p∗ (κ) > pmin (κ), the determination that contains the
“physical” branch p ∈ [p∗ (κ), +∞) is p+ (s, κ), hence we retain for s ≥ s(p∗ (κ), κ)
p
1 s
p(s, κ) = s−1 (s, κ) =
+
κ − 4 + (4 − κ)2 + 16κs
(40)
2 16
s κ −1
+ U (s),
=
2 8 κ
p
1 −1
2
Uκ (s) :=
κ − 4 + (4 − κ) + 16κs .
(41)
2κ
These expressions then lead to the following striking observation:
Remark 1.4. The same expression (40) appeared earlier in the set of tip multifractal
exponents x(1 ∧ n) in Ref. [19] [Eq. (12.19)], and is identical (for n = s) to
λκ (1 ∧ n) := x(1 ∧ n) − x1 , x1 := (6 − κ)(κ − 2)/8κ [[19], Eq. (12.37)]. The bulk
critical exponent x(1∧n) corresponds geometrically to the extremity of an SLEκ path
avoiding a packet of n independent Brownian motions diffusing away from its tip,
while x1 is the bulk exponent of the SLEκ single extremity. These local tip exponents
differ from the ones associated to the SLE tip multifractal spectrum (29)-(30) of
Refs. [4, 33, 36]. Eq. (40) for p(s, κ) is also identical to the so-called derivative
exponent ν(b, κ) (for b = s), obtained for radial SLEκ in Ref. [42], Eq. (3.1).
The exponents x(1 ∧ n) were calculated using the so-called quantum gravity
method in [17, 18, 19]. The function Uκ−1 (41) appears there as the inverse of the socalled Kniznik-Polyakov-Zamolodchikov (KPZ) relation [39] (see also Refs. [11, 14]),
which was recently proven rigorously in a probabilistic framework [25, 24, 26]. (See
also Ref. [58].) Here it maps a critical exponent in the complex (half-)plane H,
n(= s), corresponding to the boundary scaling behavior of a packet of n independent
Brownian motions, to its quantum gravity counterpart on a random surface coupled
to SLEκ , Uκ−1 (s).
13
The derivative exponents ν(b, κ) = x(1∧b)−x1 also describe the scaling behavior
of the moments of order b(= s) of the modulus of the derivative of the forward radial
SLEκ map gt in D at large time t [42]. (See also Refs. [41, 60].) In section 4.4, we
give a heuristic explanation of why the inverse function p(s, κ) = s−1 (s, κ) of the
packing spectrum s(p, κ) (39) of the unbounded whole-plane SLE coı̈ncides with the
derivative exponents ν(s, κ) of radial SLE.
Recall that p(s, κ) is analytically defined only for s ≥ smin (κ) := −(4 − κ)2 /16κ.
For κ ≤ 4, one has p(s, κ) ≥ 0 for s ≥ 0. For κ ≥ 4, p ∈ [pmin (κ), +∞], with
pmin (κ) = p(smin (κ), κ) = (κ2 − 16)/32κ. For s = 0, p(0, κ) = (κ − 4 + |κ − 4|)/16,
hence p(0, κ) = 0 for κ ≤ 4, and p(0, κ) = (κ − 4)/8 for κ ≥ 4.
Consider now the m-fold version of the unbounded inner whole-plane SLE. For
1 ≤ m ≤ 3, ∀κ, or for m ≥ 4 and κ ≤ κm (258), we have for p ≥ p∗m (κ) (20) the
average integral means and packing spectra
r
1
1 1
p
(42)
p− −
1 + 2κ ,
βm (p, κ) =
2+
m
2 2
m
sm (p, κ) := βm (p, κ) − p + 1
r
p
1 1
p
= 2 + −
1 + 2κ
(43)
m 2 2
m
p = s
,κ .
(44)
m
The inverse function, pm (s, κ) := s−1
m (s, κ), is therefore simply
∗
pm (s, κ) = s−1
m (s, κ) = m p(s, κ), s ∈ [s(pm (κ)), ∞),
(45)
where p(s, κ) is the inverse function (40) of s(p, κ) for m = 1.
For m ≥ 4 and κ ≥ κm , we have the successive integral means and packing
spectra
βm (p, κ) =
sm (p, κ) =
=
βm (p, κ) =
sm (p, κ) =
=
(4 + κ)2
3(4 + κ)2
κ2 − 4
∗
p−
, p0 (κ) =
≤p≤m
= p∗∗
m (κ),
16κ
32κ
32κ
βm (p, κ) − p + 1
(4 − κ)2
−
= smin (κ);
r
16κ 1 1
p
κ2 − 4
1
2+
p− −
1 + 2κ , p ≥ m
,
m
2 2
m
32κ
βm (p, κ) − p + 1
r
p p
1 1
p
2 + −
1 + 2κ = s
,κ .
m 2 2
m
m
14
(46)
(47)
(48)
(49)
Observe that p∗∗
m (κ) = m pmin (κ) = m p(smin (κ), κ), so that the inverse function of
−1
sm (p, κ), sm (s, κ), is now defined in the whole range s ≥ smin (κ), and is given by
pm (s, κ) = s−1
m (s, κ) = m p(s, κ) ∈ [mpmin (κ), ∞), s ∈ [smin (κ), ∞). (50)
2
2.1
2.1.1
Coefficient estimates
Computation of an and E(|an |2 ) for small n
Loewner’s method
In this paragraph we perform computations for general Loewner-Lévy processes.
Let us recall that
X
ft (z) = et z +
an (t)z n .
(51)
n≥2
By expanding both sides of Loewner’s equation (2) as power series, and identifying
coefficients, leads one to the set of equations
ȧn (t) − (n − 1)an (t) = 2
n−1
X
p
(n − p)an−p (t)λ̄ (t) = 2
p=1
n−1
X
kak (t)λ̄n−k (t), n ≥ 2; (52)
k=1
where a1 = 1; the dot means a t-derivative, and λ̄(t) = 1/λ(t). Specifying for
n = 2, 3 gives
ȧ2 − a2 = 2λ̄,
ȧ3 − 2a3 = 4a2 λ̄ + 2λ̄2 .
(53)
(54)
The first differential equation (53) (together with the uniform bound, ∀t ≥ 0, |a2 (t)| ≤
C2 < +∞; see Remark 5.2), yields
Z +∞
t
a2 (t) = −2e
e−s λ̄(s)ds.
(55)
t
In a similar way, the second one (54) leads to
Z
Z +∞
−2s
2t
2t
a3 (t) = −4e
e a2 (s)λ̄(s)ds − 2e
t
+∞
e−2s λ̄2 (s)ds,
t
R∞
The first integral invoves t u2 (s)u̇2 (s)ds = −u22 (t)/2, where u2 (s) := e−s a2 (s).
The formula for a3 then reduces to
Z +∞
2
Z +∞
2t
−s
2t
a3 (t) = 4e
e λ̄(s)ds − 2e
e−2s λ̄2 (s)ds.
(56)
t
t
15
2.1.2
Quadratic coefficients
Proposition 2.1. For Lévy-Loewner processes, we have, setting here a2 := a2 (0),
2
E(|a2 | ) = <
4
1 + η1
.
Proof - From (55), we write
∞
Z
2
−s+iLs
|a2 | = 2
e
Z
−s
∞
e
e
0
∞
Z
0
−s+iLs
ds ds + 2
Z
e
s
∞
2
−s0 −iLs0
e
0
Z
∞
Z
0
−s0 −i(Ls0 −Ls )
Z
0
s
0
e−s −iLs0 ds0 ds =
0
∞
ds ds + 2
s
−s
s
Z
0
0
0
e−s +i(Ls −Ls ) ds0 ds.
e
0
Using now the characteristic function (4) for Ls − Ls0 , and taking care of the relative
order of s and s0 , we get
2
∞
Z
E(|a2 | ) = 2
e
0
−s
Z
∞
e
−s0 −(s0 −s)η1
0
Z
ds ds + 2
s
∞
−s
Z
e
0
s
0
0
e−s −(s−s )η1 ds0 ds,
0
and the result follows.
For calculations involving the third order term a3 as given by (56), and in order
to avoid repetitions, we have computed at once E(|a3 − µa22 |2 ), where µ is a real
constant. The detail of the calculation is given in Appendix in Section 5.2.1. Let
us state here the result.
Proposition 2.2. If µ is a real coefficient then
E(|a3 − µa22 |2 ) =
16(1 − µ)2 (4 + η2 )
16(1 − µ)(2 + η1 )
2
8(1 − µ)(1 − 2µ)
<
−
+
.
+
(1 + η1 )(2 + η2 )(3 + η1 ) (1 + η1 )(2 + η2 )(3 + η1 ) 2 + η2
(η1 + 1)(η1 + 3)
In the real η case:
E(|a3 − µa22 |2 ) =
32(1 − µ)2 (3 + η2 ) − 8(1 − µ)(6 + 2η1 + η2 ) + 2(1 + η1 )(3 + η1 )
.
(1 + η1 )(2 + η2 )(3 + η1 )
In the SLE case (i.e., for η` = κ2 `2 ):
E(|a3 −
µa22 |2 )
(108 − 288µ + 192µ2 ) + (88 − 208µ + 128µ2 )κ + κ2
=
.
(1 + κ)(2 + κ)(6 + κ)
16
2.1.3
Some corollaries
The first one gives, for µ = 0, the analogue of Loewner’s estimate.
Corollary 2.1. For Lévy-Loewner processes with η real, we have
(η1 − 1)(η1 − 3)
1
2
24 + 2
.
E(|a3 | ) =
(1 + η1 )(3 + η1 )
2 + η2
In the SLE case:
E(|a3 |2 ) =
(57)
108 + 88κ + κ2
.
(1 + κ)(2 + κ)(6 + κ)
Notice the speciel role played by η1 = 1, 3, corresponding to κ = 2, 6: the result no
longer depends on η2 , and equals, respectively, 3 and 1.
The second corollary shows that there is no Fekete-Szegö counter-example in
the SLE family. To f := f0 , an SLEp
κ whole-plane map, we associate its oddified
function as above, that is h0 (z) = z f (z 2 )/z 2 = z + b3 z 3 + b5 z 5 + · · · An easy
computation gives b5 = 12 a3 − 14 a22 . Setting µ = 14 in the above proposition gives
18 + 9η2 − 4η1 + 2η12
3
1
2
E(|b5 | ) = <
+
.
4(1 + η1 )(2 + η2 )(3 + η1 ) 4 (1 + η1 )(3 + η1 )
In the case of a real η,
6 + 3η2 − η1 + η12 /2
12 + 44κ + κ2
E(|b5 | ) =
=
,
(1 + η1 )(3 + η1 )(2 + η2 )
(1 + κ)(2 + κ)(6 + κ)
2
where the last expression has been specified for the SLE case, and is always less
than or equal to 1 (equality holding only for κ = 0).
Consider the Schwarzian derivative, S(z) := f [3] (z)/f 0 (z)−(3/2)(f 0[2] (z)/f 0 (z))2 .
One obtains S(0) = 6(a3 − a22 ), corresponding to µ = 1, and giving the expected
values E[|S(0)|2 ] = 72/(2 + η2 ), and for SLE, E[|S(0)|2 ] = 36/(1 + κ).
A few comments are in order here:
– We noticed that E(|a2 |2 ) = E(|a3 |2 ) = 1 for κ = 6. We return to this in the next
sections after performing some computer experiments.
– For all values of κ, E(|b5 |2 ) ≤ 1: there is no Fekete-Szegö counterexample in the
SLE-family. Using the Schoenberg property of the Lévy symbol η [2], it can also
be seen that there is no counterexample in expectation for a general Lévy-Loewner
process with real η. The question remains open for higher order terms or higher
moments; this will be studied elsewhere.
– It is known that |S(0)| ≤ 6 whenever f is injective. Conversely, if (1−|z|2 )2 |S(z)| ≤
2, then f is injective; in our case the boundary value 2 is reached for κ ≥ 8.
17
2.1.4
Next order
The quadratic expectation of the next order coefficient, E(|a24 |), can still be computed by hand, which yields
E |a4 |2
=
4!23
(η1 + 1)(η1 + 3)(η1 + 5)
4(η1 − 1)(η1 − 3)η2 (η2 − 4)(η1 + 3)
+
,
3(η1 + 1)(η1 + 3)(η1 + 5)(η2 + 2)(η2 + 4)(η3 + 3)
(58)
(59)
and for SLE,
8 κ5 + 104κ4 + 4576κ3 + 18288κ2 + 22896κ + 8640
E |a4 |2 =
.
9
(κ + 10)(3κ + 2)(κ + 6)(κ + 1)(κ + 2)2
The results (57) and (58) obtained so far for E(|an |2 ), n = 3, 4, call for the
following observations.
Remarks 2.1. –After the first term in the expression for E(|an |2 ), one notes the
presence in numerators of the common factors (η1 − 1)(η1 − 3), thus vanishing for
η1 = 1 or 3. For η1 = 3 (or κ = 6), the first term itself, thus E(|an |2 ), equals 1; for
η1 = 1 (or κ = 2) it equals n.
–Somehow surprisingly, all the coefficients of the polynomial expansions in κ are
positive.
–For κ → ∞ (or η → ∞), these expectations vanish as κ−1 .
All these patterns will be confirmed at higher orders, to which we now turn.
2.2
Computational experiments
As one may see, these computations become more and more involved. Moreover,
its seems difficult to find a closed formula for all terms. This section is devoted to
the description of an algorithm that we have implemented on matlab to compute
E(|an |2 ). This algorithm is divided into two parts: the first encodes the computation
of an , while the second uses it to compute E(|an |2 ). Since the important cases of
SLE and α-stable processes both have real Lévy symbols η, we restrict the study to
the latter case.
For the encoding of an , we observe that they are linear combinations of successive
integrals of the form
Z ∞
Z ∞
Z ∞
−iα1 Ls1 −β1 s1
−iα2 Ls2 −β2 s2
ds1 e
ds2 e
...
dsk e−iαk Lsk −βk sk .
(60)
t
s1
sk−1
Their expectations are encoded as
(α1 , β1 ) . . . (αk , βk ) (1 ≤ k ≤ n),
18
(61)
Figure 2: Graphs of the SLEκ map κ 7→ E(|an |2 ) for n = 1, · · · , 8.
and are explicitly computed by using as above the strong Markov property and the
Lévy characteristic function (4):
(α1 , β1 ) . . . (αk , βk ) =
k−1
Y
[βk + βk−1 + . . . + βk−j + η(αk + αk−1 + . . . + αk−j )]−1 .
j=0
Next, in order to compute |an |2 , we need to evaluate the expectation of products of
integrals such as (60) with complex conjugate of others, that we symbolically denote
by
[(α1 , β1 ) . . . (αk , βk ); (−α10 , β10 ) . . . (−α`0 , β`0 )] (1 ≤ k, ` ≤ n).
(62)
The product integrals may be written as a sum of ( k+`
k ) ordered integrals with k + `
variables: the k first ones and the ` last ones are ordered and the number of ordered
integrals corresponds to the number of ways of shuffling k cards in the left hand with
` cards in the right hand. This sum is quite large and, in order to systematically
compute it, we write its expectation as the sum of expectations of integrals of the
form (61) that begin with a term of type (α1 , β1 ) or with a term of type (−α10 , β10 ),
thus reducing the work to a computation at lower order.
Using dynamic programing, we performed computations (formal up to n = 8 and
numerical up to n = 19) on a usual computer. The results are reported in Appendix
B, section 5.2.2. They fully confirm the validity of Remarks 2.1.
The graphs given in Figure 2 for the SLEκ map κ 7→ E(|an |2 ), for n = 1, · · · , 8,
19
Figure 3: Zooming in near κ = 6.
illustrate the phenomena described above, in particular the striking constant value
E(|a2n |) = 1 for κ = 6 (Fig. 3).
3
Theorems and Proofs
3.1
Expected conformal maps for Lévy-Loewner evolutions
3.1.1
Expectation of f0 (z)
In this first section, we give an explicit expression for the expectations of the coefficients an (t) of the expansion (51) in the Lévy-Loewner setting, thereby obtaining
the expectation of the map, E[f0 (z)], and of its derivative.
The differential recursion (52) in Section 2 then becomes, for λ(t) := eiLt , and
in terms of the auxiliary function un (t),
un (t) := an (t)e−(n−1)t
n−1
X
u̇n (t) = 2
kXtn−k uk (t),
(63)
(64)
k=1
where Xt is defined as
Xt := e−t−iLt .
20
(65)
The recursion (64) can be rewritten under the simpler form:
u̇n = Xt [u̇n−1 + 2(n − 1)un−1 ].
(66)
Recall that u1 = a1 = 1, while the next term of this recursion, as already seen in
Eqs. (55), is
Z +∞
u2 (t) = −2
dsXs .
(67)
t
Similarly, we can write the general solution un , for n ≥ 2, under the form
Z +∞
un (t) = −2
dsXs vn (s),
(68)
t
with v2 (s) = 1, and rewrite the differential equation (66) as an integral equation
Z +∞
dsXs vn−1 (s).
(69)
vn (t) = Xt vn−1 (t) − 2(n − 1)
t
Define then the multiplicative and integral operators X and J such that
X v(t) := Xt v(t),
Z +∞
J v(t) := −2
dsXs v(s).
(70)
(71)
t
The solution to (67), (68) and (69) can then be written as the operator product
un = J ◦ [X + (n − 1)J ] ◦ · · · ◦ (X + 2J )1
n−2
Y
◦(X + (k + 1)J )1,
= J
(72)
k=1
where 1(= v2 ) is the constant function equal to 1 on R+ .
Next, recall the strong Markov property of the Lévy process, which implies the
(law)
identity in law: ∀s ≥ t, Ls = Lt + L̃s−t , where L̃s0 is an independent copy of the
Lévy process, also started at L̃0 = 0. Therefore, the process Xt (65) is, in law,
(law)
Xs = Xt X̃s−t , ∀s ≥ t
(73)
0
where X̃s0 := e−s −L̃s0 , s0 ≥ 0, is an independent copy of that process, with X̃0 = 1.
The operator J (71) can then be written as
Z +∞
(law)
J v(t) = −2Xt
dsX̃s v(s + t)
(74)
0
=
X ◦ J˜v(t),
21
(75)
R +∞
with J˜v(t) := −2 0 dsX̃s v(s + t). By iteration of the use of the Markov property,
Eq. (72) can be rewritten as
(law)
un = J ◦ X 1 + (n − 2)J˜[n−1] ◦ · · · ◦ X 1 + 2J˜[1] 1
(law)
n−2
Y
J
=
◦[X 1 + (k + 1)J˜[k] 1,
(76)
k=1
where the integral operators J˜[k] , k = 1, · · · , n − 2, involve successive independent
[k]
copies, X̃sk , k = 1, · · · , n − 2, of the original exponential Lévy process Xs . We
therefore arrive at the following explicit representation of the solution (72)
Z +∞
Z +∞
n−2
Y
(law)
n−1
[k] k
1 − 2(k + 1)
dsk X̃sk
.
(77)
un (t) = −2
dsXs
t
0
k=1
As mentioned in the introduction, the conjugate whole-plane Lévy-Loewner evolution e−iLt ft eiLt z should have the same law as f0 (z). At order n, we are thus
interested in the stochastically rotated coefficients:
ei(n−1)Lt an (t) = (Xt )−(n−1) un (t).
Using again the identity in law (73) in (77), we arrive at
Z +∞
Z
n−2
Y
(law)
n−1
i(n−1)Lt
dsX̃s
e
an (t) = −2
1 − 2(k + 1)
0
(law)
=
0
k=1
+∞
k
dsk X̃s[k]
k
(78)
an (0),
which, as it must, no longer depends of t.
All factors in (78) involve successive independent copies of the Lévy process, and
their expectations can now be taken independently. Recalling the form (4) of the
Lévy characteristic function, we have E[(X̃s )k ] = e−(ηk +k)s . Thus
Z +∞
Z +∞
n−2
Y
[k] k n−1
E[an (0)] = −2
ds E[X̃s ]
1 − 2(k + 1)
dsk E X̃sk
0
= −2
k=1
n−2
Y
1
2(k + 1)
1−
ηn−1 + n − 1 k=1
ηk + k
0
.
(79)
We finally obtain:
Theorem 3.1. For n ≥ 2, setting an := an (0),
an (0)
E(an )
(law)
=
=
ei(n−1)Lt an (t),
Qn−2
n−2
Y ηk − k − 2
k=1 (ηk − k − 2)
−2 Q
=
.
n−1
η
+
k
+
1
(η
+
k)
k+1
k
k=1
k=0
22
(80)
(81)
Corollary 3.1. The expected conformal map E[f0 (z)] of the whole-plane LévyLoewner evolution, in the setting of Theorem 1.1, is polynomial of degree k + 1
if there exists a positive k such that ηk = k + 2, has radius of convergence 1 for
an α-stable Lévy process of symbol ηn = κnα /2, α ∈ (0, 2], except for the Cauchy
process α = 1, κ = 2, where E[f0 (z)] = ze−z .
Proof. From Theorem 3.1, E[f0 (z)] is polynomial if there exists k ∈ N such that
ηk = k + 2, as all E(an ) then vanish for n ≥ k + 2. Otherwise, use D’Alembert’s
criterion, applied here to
|ηn−1 − n − 1|
|E(an+1 )|
= lim
= 1,
n→∞ |E(an )|
n→∞
|ηn + n|
lim
for an α-stable symbol, ηn = κ|n|α /2, ∀α ∈ (0, 2], except if α = 1 and κ = 2, for
which the limit vanishes. In that case, Eq. (81) gives E(an ) = (−1)n−1 /(n − 1)! for
n ≥ 2, thus E[f0 (z)] = ze−z and E[f00 (z)] = (1 − z)e−z .
3.1.2
Expectations for the odd map h0 (z)
p
The oddified map ht (z) := z ft (z 2 )/z 2 obeys the Loewner equation
z
λ(t) + z 2
ḣt (z) = h0t (z)
.
2
λ(t) − z 2
(82)
Its series expansion
e−t/2 ht (z) = z +
X
b2n+1 (t)z 2n+1
(83)
n≥1
gives the recursion: ḃ2n+1 = nb2n+1 +
transformed into the set of equations
Pn−1
k=0
λ̄n−k (2k + 1)b2k+1 , with b1 = 1. This is
wn (t) := b2n+1 (t)e−nt , w0 = 1,
n−1
X
ẇn (t) =
(2k + 1)Xtn−k wk (t)
(84)
k=0
= (2n − 1)Xt wn−1 (t) + Xt ẇn−1 (t).
(85)
The last equation is similar to Eq. (66), except for 2n−1 here replacing 2n−2 there,
and an index shifted boundary condition w0 = 1 replacing u1 = 1. Its solution can
thus be written, as in (72), as the operator product
1 3 1
J ◦ X + n − J ◦ ··· ◦ X + J 1
2
2
2
n−1
Y
1
1
=
J
◦ X + k + J 1,
2 k=1
2
wn =
23
(86)
where 1(= w0 ) is the constant function equal to 1 on R+ .
As mentioned in the introduction,
the conjugate odd whole-plane Lévy-Loewner
−(i/2)Lt
(i/2)Lt
evolution e
ht e
z should have the same law as h0 (z). At order n, we
are thus interested in the stochastically rotated coefficients:
einLt b2n+1 (t) = (Xt )−n wn (t).
Comparing (86) here to (72) above, and adapting from the general formula (78), we
arrive directly at the final identity in law for the odd coefficients
inLt
e
b2n+1 (t)
+∞
Z
(law)
dsX̃sn
−
=
0
(law)
=
n−1
Y
Z
+∞
1 − (2k + 1)
0
k=1
k
dsk X̃s[k]
k
(87)
b2n+1 (0),
which, as it must, no longer depends of t. Again, all factors in (87) involve successive independent copies of the Lévy process, whose expectations can be taken
independently. Thus
+∞
Z
ds E[X̃sn ]
E[b2n+1 (0)] = −
0
n−1
Y
Z
1 − 2(k + 1)
2k + 1)
1
= −
1−
ηn + n k=1
ηk + k
dsk E
0
k=1
n−1
Y
+∞
k X̃s[k]
k
.
(88)
We finally obtain for the odd whole-plane Lévy-Loewner evolution:
Theorem 3.2. For n ≥ 1, setting b2n+1 := b2n+1 (0),
b2n+1 (0)
E(b2n+1 )
(law)
=
=
einLt b2n+1 (t),
Qn−1
n−1
(ηk − k − 1) Y ηk − k − 1
k=1
− Qn
=
.
ηk+1 + k + 1
k=1 (ηk + k)
k=0
(89)
(90)
Corollary 3.2. The expected conformal map E[h0 (z)] of the oddified whole-plane
Lévy-Loewner evolution, in the setting of Theorem 1.1, is polynomial of degree 2k +1
if there exists a positive k such that ηk = k +1, has radius of convergence 1 for an αstable Lévy process of symbol ηn = κnα /2, α ∈ (0, 2], except for the Cauchy process
2
α = 1, κ = 2, where E[h0 (z)] = ze−z /2 .
24
3.1.3
Some results for Lévy-Loewner maps
In the general case (or in the specific case of SLEκ ), the formula (81) gives for the
first terms:
4
2
=−
E(a2 ) = −
η1 + 1
2+κ
η1 − 3 2
κ−6
E(a3 ) = −
=−
η2 + 2 η1 + 1
(1 + κ)(2 + κ)
η2 − 4 η1 − 3 2
4(κ − 2)(κ − 6)
E(a4 ) = −
=−
.
η3 + 3 η2 + 2 η1 + 1
(6 + 9κ)(1 + κ)(2 + κ)
For the oddified map, (90) gives
2
1
=−
E(b3 ) = −
η1 + 1
κ+2
η1 − 2
κ−4
E(b5 ) = −
=−
(η1 + 1)(η2 + 2)
2(κ + 1)(κ + 2)
(κ − 4)(2κ − 3)
(η1 − 2)(η2 − 3)
=−
.
E(b7 ) = −
(η1 + 1)(η2 + 2)(η3 + 3)
3(κ + 1)(κ + 2)(3κ + 2)
An interesting further identity, valid for n ≥ 1, gives the truncated series
1
Sn := 1 + 2 E(a2 ) + · · · + n E(an ) = − ηn−1 − (n + 1) E(an )
2
1
= − ηn + n E(an+1 ).
(91)
2
Due to the peculiar factorized and recursive form of E(an ) in theorem 3.1 (respectively, of E(bn ) in theorem 3.2), we have seen in corollary 3.1 (respectively, 3.2) that
if there exits an integer N such that ηN = N + 2 (respectively, ηN = N + 1), E[f0 (z)]
is polynomial of degree N +1 (respectively, E[h0 (z)] is polynomial of degree 2N +1).
In the first case, E(aN +` ) = 0, ∀` ≥ 2, and E[f00 (z = 1)] = SN +1 = 0, therefore
the derivative E[f00 (z)] necessarily contains the monomial (1 − z) as a factor.
The first such case, N = 1, gives a Lévy symbol η1 = 3. This includes in
particular the SLEκ process for κ = 6 (recall then that ηn = κn2 /2), for which
1
E[f0 (z)] = z − z 2 /2 = 1 − (1 − z)2 ,
2
E[f00 (z)] = 1 − z.
(92)
The N = 2 case gives η2 = 4. This includes in particular the SLEκ process for
κ = 2, for which
1
E[f0 (z)] = z − z 2 + z 3 /3 = 1 − (1 − z)3 ,
3
E[f00 (z)] = (1 − z)2 .
(93)
25
More generally, the SLEκ expected map, z 7→ E[ft (z)], is polynomial for the decreasing sequence of values κ = κN := 2(NN+2)
, N ≥ 1.
2
For the oddified Lévy-Loewner evolution, the N = 1 first case gives η1 = 2. This
includes in particular the SLEκ for κ = 4, for which
E[h00 (z)] = 1 − z 2 .
(94)
The odd SLEκ expected map E[ht (z)] is polynomial for κ = κ̃N :=
3.2
2(N +1)
,
N2
N ≥ 1.
Derivative moments
In this section, we prove the following result, motivated by the observations made
in sections 2 and 2.2:
Theorem 3.3. Let (ft )t≥0 be the Loewner whole-plane process driven by the Lévy
process Lt with Lévy symbol η. We write
X
ft (z) = et z +
an (t)z n ,
n≥2
and an = an (0). We also consider the oddification of ft ,
X
p
b2n+1 (t)z 2n+1
ht (z) = z ft (z 2 )/z 2 = et/2 z +
n≥1
and set b2n+1 = b2n+1 (0).Then:
(i) If η1 = 3, we have
E(|an |2 ) = 1, ∀n ≥ 1;
this case covers SLE6 .
(ii) If η1 = 1, η2 = 4, we have
E(|an |2 ) = n, ∀n ≥ 1;
this case covers SLE2 .
(iii) If η1 = 2, we have
E(|b2n+1 |2 ) =
this case covers SLE4 .
26
1
, ∀n ≥ 1;
2n + 1
3.2.1
The Beliaev-Smirnov equation
In their study of the harmonic measure for SLE, Beliaev and Smirnov (BS) [4] first
consider a standard radial (outer) SLE process (gt (z), t ≥ 0), from C \ Kt to the
complement D− of the unit disk, where, as usual, Kt denotes the SLE hull at time
t. This SLE process satisfies a standard ODE, which can be continued to negative
times (via√ a two-sided Brownian motion Bt in the Schramm-Loewner source term
λ(t) = ei κBt ). The harmonic spectrum is best studied via the inverse map, gt−1 ,
which satisfies a Loewner-type PDE (as the whole-plane evolution considered in this
article).
Since the processes gt−1 and g−t have the same law (up to conjugation by
√
ei κBt ), BS redefine in Ref. [4] a radial SLE as f˜t := g−t (denoted there by ft ), thus
mapping D− to C \ K−t . Then they show that the expectation
F̃ (z, t) := E(|f˜t0 (z)|p ),
(95)
where p is real, is solution to the differential equation
r(r2 − 1)
r4 + 4r2 (1 − r cos θ) − 1
F̃
+
F̃r
(r2 − 2r cos θ + 1)2
r2 − 2r cos θ + 1
2r sin θ
− 2
F̃θ + ΛF̃ − F̃t = 0,
r − 2r cos θ + 1
p
(96)
with z = reiθ , and where subscripts represent partial derivatives of F̃ with respect
to r, θ and t, and where Λ stands for the infinitesimal generator of the SLE driving
Brownian process, i.e., Λ = (κ/2)∂ 2 /∂θ2 .
To derive (96), they consider the martingale Ms := E(|f˜t (z)0 |p | Fs ), where Fs
denotes the σ−algebra generated by {Bτ ; τ ≤ s}. From the SLE Markov property,
they show (Lemma 2 in [4]) that
E(|f˜t0 (z)|p | Fs ) = |f˜s0 (z)|p F̃ (zs , t − s),
√
in terms of the conjugate variable zs := f˜s (z)e−i κBs . Expressing the fact that the ds
drift term vanishes in the Itô derivative of the right-hand side, then gives equation
(96) above.
The next step in their derivation is to remark that, by stationarity, the limit of
−t ˜
e ft (z) as t → +∞ exists, and has the same law as the value fˆ0 (z) at proper time
zero of the (outer) whole-plane SLE (denoted by F0 (z) in [4]). Rewrite F̃ (z, t) (95)
above trivially as
F̃ (z, t) = eσpt E(|e−σt ft0 (z)|p ), (σ = 1),
(97)
to obtain
F (z) := E[|fˆ00 (z)|p ] = lim e−σpt F̃ (z, t).
t→∞
(98)
Note that this exterior whole-plane map fˆ0 (z), acting on the exterior D− of the
unit disk, has precisely the same law as the conjugate via z 7→ 1/z of the interior
27
whole-plane SLE map f0 (z) that we consider in this article. Substituting (97) into
(96) and taking the large t limit, BS thereby obtain the following equation for F (z)
4
r(r2 − 1)
r + 4r2 (1 − r cos θ) − 1
Fr
−
σ
F
+
p
(r2 − 2r cos θ + 1)2
r2 − 2r cos θ + 1
2r sin θ
− 2
Fθ + ΛF = 0.
(99)
r − 2r cos θ + 1
For our interior case, we similarly introduce the function f˜t , t ≥ 0, as the continuation g−t of the standard inner radial SLE process gt to negative times, which has
the same law as its inverse map gt−1 . Then the limit et f˜t (z) as t → +∞ exists, and
has the same law as the inner whole-plane process f0 (z) considered in this work.
This amounts to formally taking σ = −1 in (97), in effect changing the sign of the
term −σpF in (99) that results from the time-derivative term in (96). This simple
observation results in:
Proposition 3.1. For the interior whole-plane Schramm-Loewner evolution, as considered here in the setting of Theorem 1.1, the expected moments of the derivative modulus, F (z) = E(|f00 (z)|p ), satisfy the Beliaev-Smirnov equation (99) with
σ = −1, and Λ = (κ/2)∂ 2 /∂θ2 the generator of the driving Brownian process.
Finally, note that the BS derivation for SLE, as recalled above, is also valid for the
Lévy-Loewner evolution, which possesses the same Markov property, together with
the existence of similar whole-plane stationary limits. Stochastic calculus (and Itô
formula) can be generalized to Lévy processes [2], resulting in the same martingale
argument. As mentioned by Beliaev and Smirnov in [4], one simply has to take for
Λ in (99) the generator of the driving Lévy process. We therefore state:
Proposition 3.2. For the interior whole-plane Lévy-Loewner evolution, as considered here in the setting of Theorem 1.1, the expected moments of the derivative modulus, F (z) = E(|f00 (z)|p ), satisfy the Beliaev-Smirnov equation (99) with σ = −1,
and Λ the generator of the driving Lévy process.
3.2.2
Whole-plane SLE solutions
We first study the SLEκ case. Let us switch to z, z variables, instead of polar
coordinates, and write F (z) above as
F (z, z̄) := E(|f00 (z)|p ) = E[(f00 (z))p/2 (f¯00 (z̄))p/2 ].
(100)
Using ∂ := ∂z , ∂ := ∂z , Eq. (99) then becomes for σ = −1
κ
z+1
z+1
1
1
2
z∂F +
z∂F −p
+
− 2 F = 0. (101)
− (z∂−z∂) F +
2
z−1
z−1
(z − 1)2 (z − 1)2
To solve this equation, we shall need in the sequel three little lemmas.
28
Lemma 3.1. The space of formal series in z, z that are solutions of (101) is onedimensional.
Proof - The Lemma is an easy consequence of the two following observations:
First, the differential operator, P(D), involved in (101), is polynomial in z∂ and z̄∂,
and the monomials z k z ` are eigenvectors of the latter two operators.
Second, the function of z and z̄, factor of F in P(D), may be written as A(z)+A(z̄),
with A(0) = 0.
Now, assuming that G is a solution of (101) with G(0, 0) = 0, it suffices to prove
that, necessarily, G = 0. We argue by contradiction: If not, consider the minimal
(necessarily non constant) term ak,` z k z̄ ` in the series, with ak,` 6= 0 and k+` minimal
(and non vanishing). Then P(D)[F ] will have a minimal non-vanishing term of the
form
κ
−ak,` (k − `)2 + k + ` z k z̄ ` ,
2
contradicting the fact that P(D)[F ] = 0.
Lemma 3.2. The quantity F (z, z̄ = 0) satisfies the boundary equation obtained by
setting z̄ = 0 in (101):
κ
z+1
1
2
P(∂)[F (z, 0)] := − (z∂) +
z∂ − p
− 1 F (z, 0) = 0. (102)
2
z−1
(z − 1)2
The complex conjugate equation also holds for F (z = 0, z̄).
Proof - The
P bi-analytic function F (z, z̄) has a double series expansion of the form
F (z, z̄) = k≥0,`≥0 ak,l z k z̄ ` . When acting on it with a differential operator P(D) as
in (101), the resulting double series must vanish identically, hence all its coefficients
ak,` must as well. This implies that the variables z and z̄ can be considered as
two independent complex variables in (101). By symmetry, the complex conjugate
equation also holds for F (z = 0, z̄).
Lemma 3.3. The action of the operator P(D) (101) on a function of the factorized
form F (z, z̄) = ϕ(z)ϕ(z̄)P (z, z̄) is, by Leibniz’s rule, given by
κ
ϕϕ(z∂ − z∂)2 P − κ(z∂ − z∂)(ϕϕ)(z∂ − z∂)P
2
z+1
z+1
+ κ(z∂ϕ)(z∂ϕ)P + ϕϕ
z∂P + ϕϕ
z∂P
z−1
z−1
κ
κ
z+1
z+1
2
2
+ − ϕ(z∂) ϕ − ϕ(z∂) ϕ + ϕ
z∂ ϕ P
z∂ϕ + ϕ
2
2
z−1
z−1
1
1
− p
+
− 2 ϕϕP.
(103)
(z − 1)2 (z − 1)2
P(D)[ϕϕ̄P ] = −
29
Corollary 3.3. For the the particular choice: P (z, z̄) := P (z z̄), the first line of
(103) identically vanishes.
Proof - P is then radial, and the differential operator in the first line acts on the
angular variable only.
Corollary 3.4. For the particular choice:
ϕ(z) = F (z, z̄ = 0), ϕ̄(z̄) = F (z = 0, z̄),
(104)
the last two lines of (103) in Lemma 3.3 vanish identically.
Proof - Use Lemma 3.2. The last two lines of (103) are precisely of the form
P × ϕ̄P(∂)ϕ + ϕ P(∂)ϕ̄ = 0.
Corollary 3.5. The function
F (z, z̄) = E |f00 (z)|p = E (f00 (z))p/2 (f00 (z̄))p/2
is the unique solution of (101) such that F (0, 0) = 1. The function
ϕ(z) = F (z, 0) = E (f00 (z))p/2
is the unique solution of (102) such that F (z = 0, 0) = 1. It satisfies corollary 3.4.
In the particular case p = 2, and for SLEκ , with κ = 6 or 2, we have obtained
above the derivative expectations (92) and (93):
ϕ(z) = E f00 (z) = (1 − z)α , α = 1, κ = 6; α = 2, κ = 2.
(105)
From Corollary 3.5, we know that they are annihilated by the boundary operator
P(∂) of Lemma 3.2 and that the last two lines of (103) identically vanish.
Denote then by Psing (D) the singular operator on the second line of (103), that
contains the pole at z = 1, z̄ = 1. Its action, for ϕα (z) := (1 − z)α , gives the
homogeneous form
z+1
z+1
κα2 z z̄
+
z∂ +
z∂ P.
(106)
Psing (D)[P ] = ϕα ϕ̄α
(1 − z)(1 − z̄) z − 1
z−1
Thanks to Lemma 3.3, it is now natural to look for radial solutions, P (z, z̄) =
P (z z̄), that make (106) vanish. The resulting equation is simply
P 0 (z z̄)
κα2 1
=
,
P (z z̄)
2 1 − z z̄
30
(107)
which is immediately solved, for P (0) = 1, into
P (z z̄) = (1 − z z̄)−κα
2 /2
.
(108)
From Corollaries 3.3 and 3.4, we obtain that
F (z, z̄) =
κ
(1 − z)α (1 − z̄)α
, β = α2 ,
β
(1 − z z̄)
2
(109)
is the unique solution to the differential equation (101), such that F (0, 0) = 0, if
and only if ϕα (z) = (1 − z)α is a solution to the boundary equation (102) of Lemma
3.2. From (105), we already know this to hold true for p = 2, in the two cases
α = 1, κ = 6, or α = 2, κ = 2.
In the general case, we obtain:
P(∂)[ϕα ] = Aϕα + Bϕα−1 + Cϕα−2
(110)
κ 2
A = − α + α + p,
(111)
2
κ
α(2α − 1) − 3α,
(112)
B =
2
κ
C = − α(α − 1) + 2α − p.
(113)
2
Notice that A+B +C = 0. The boundary equation (102) P(∂)[ϕα ] = 0 thus reduces
to the set of equations: A = 0, B = 0, which is solved into
(6 + κ)(2 + κ)
6+κ
, p = p(κ) = p1 (κ) :=
.
(114)
2κ
8κ
This set of values naturally includes the above cases (105) for κ = 6 and κ = 2.
From Corollary 3.5, we therefore obtain the general result:
α = α(κ) =
Theorem 3.4. The whole-plane SLEκ map f0 (z) has derivative moments
E (f00 (z))p/2 = (1 − z)α ,
(1 − z)α (1 − z̄)α
E |f00 (z)|p =
,
(1 − z z̄)β
for the special set of exponents p = κα(α + 1)/6 = (6 + κ)(2 + κ)/8κ, with α =
(6 + κ)/2κ and β = κα2 /2 = (6 + κ)2 /8κ.
Corollary 3.6. The whole-plane SLEκ map f0 (z) has first and second derivative
moments, for κ = 6:
E(f00 (z)) = 1 − z, E(|f00 (z)|2 ) =
(1 − z)(1 − z̄)
;
(1 − z z̄)3
for κ = 2:
E(f00 (z))
2
= (1 − z) ,
E(|f00 (z)|2 )
31
(1 − z)2 (1 − z̄)2
=
.
(1 − z z̄)4
In the setting of Theorem 3.3, the coefficient n2 E(|an |2 ) is that of the term of
order (z z̄)n−2 in E(|f00 (z)|2 ). It can be obtained directly from the explicit expressions
in Corollary 3.6, as E(|an |2 ) = 1 for κ = 6, and E(|an |2 ) = n for κ = 2. Equivalently,
we can evaluate the respective integral means
Z
1
1 + z z̄
1 + 4z z̄ + (z z̄)2
E(|f00 (z)|2 ) |dz| =
;
,
2π ∂D
(1 − z z̄)3
(1 − z z̄)4
which establishes the equivalent Corollary 1.2. This achieves for SLEκ the proof of
cases (i) and (ii) of Theorem 3.3.
Remark 3.1. The expressions for the second moments in Corollary 3.6 have been
obtained in [49], as peculiar solutions to a double recursion, seemingly obtained with
the help of maple; the set of particular solutions (109), (114) is simply stated there.
3.2.3
Lévy-Loewner evolution
Theorem 3.5. If a Lévy process has its first m symbols (m ≥ 1) given by ηj =
κj 2 /2, 1 ≤ j ≤ m, with κ = 6/(2m − 1), then the associated Lévy-Loewner map
f0 (z) has the same derivative moments of order p as SLEκ , for the particular value
of the exponent p = m(m + 1)/(2m − 1), as given in Theorem 3.4 with α = m, β =
3m2 /(2m − 1).
Remark 3.2. The cases m = 1 and m = 2 give respectively the condition η1 = 3
with an equivalent SLEκ parameter κ = 6, with p = 2, α = 1, β = 3, and the
conditions η1 = 1, η2 = 4, corresponding to κ = 2, p = 2, α = 2, β = 4, as in
Corollary 3.6. Cases (i) and (ii) of Theorem 3.3 thus follow.
Proof. For the whole-plane Lévy-Loewner evolution, the Beliaev-Smirnov equation
(101) becomes [4]
1
1
z+1
z+1
z∂F +
z∂F − p
+
− 2 F = 0.
(115)
ΛF +
z−1
z−1
(z − 1)2 (z − 1)2
The action of the Lévy infinitesimal generator Λ on a term z k z̄ ` is
Λ [z k z̄ ` ] = −η(k − `)z k z̄ ` ,
where η(·) here real and even. It is such that η(0) = 0, therefore for any n ∈ Z,
Λ [(z z̄)n ] = 0.
For the set of solutions F (z, z̄) (109) of (101), as given in Theorem 3.4, we thus
have
1
ΛF (z, z̄) =
Λ[(1 − z)α (1 − z̄)α ].
(116)
(1 − z z̄)β
32
If the exponent α = (6 + κ)/2κ equals an integer m ≥ 1, ϕα = (1−z)α is polynomial
of order m, and Λ[ϕα ϕ̄α ] contains only the finite set of Lévy symbols {η1 , . . . , ηm }. If
this set coı̈ncides with the set of values (κ/2)`2 for ` = {1, · · · , m}, the action of the
Lévy generator Λ on ϕα ϕ̄α in (116) coı̈ncides with that of the Brownian generator
− κ2 (z∂ − z∂)2 . In this case, F (z, z̄) (109), solution of the SLEκ equation (101), is
also a solution of the Lévy-Loewner equation (115), and Theorem 3.4 is also valid
for a Lévy-Loewner evolution with such symbols.
3.3
Odd whole-plane SLE
p
In this section, we study the oddified whole-plane SLEκ map, ht (z) = z ft (z 2 )/z 2 ,
and derive the analogue of the Beliaev-Smirnov equation for its derivative moments,
before proceeding along lines similar to those in section 3.2.2, in order to find solutions to that equation.
3.3.1
Martingale argument
The Loewner equation for ht (z) is easily derived from the one (2) governing ft (z),
with a driving function λ(t), as
z 0 λ(t) + z 2
.
∂t ht (z) = ht (z)
2
λ(t) − z 2
(117)
To avoid cumbersome factors of 2, it is convenient in this section to work with
(t, z) 7→ h2t (z), which is a normalized whole-plane Loewner process with
∂t h2t (z) = zh02t (z)
λ(2t) + z 2
.
λ(2t) − z 2
(118)
Note that for this oddified whole-plane process, the underlying probability measure
is no longer a single Dirac measure, but the barycenter of two Dirac masses at two
diametrically opposite points, λ(2t) and −λ(2t).
In the √case where (f√t ) is the SLEκ Loewner chain, we can write, instead of
λ(2t) = ei κB2t , ξt := ei 2κBt which has the same law. We then follow the same
method as in [4], as recalled above, to find an equation satisfied by
F (z) := E(|h00 (z)|p ).
(119)
To this aim, we consider the odd whole-plane map at time 0, h0 (z), as a particular
large-time limit, limt→∞ et f˜t (z), where f˜t is now the (inner) radial Loewner process
satisfying
z 2 + ξt
∂t f˜t (z) = z f˜t0 (z) 2
.
(120)
z − ξt
33
It is easy to see that the Markov Lemma 2 in [4] goes through for f˜ in this new
setting; we can then argue as in Lemma 4 therein, namely by using a similar martingale argument. More precisely, we consider the martingale with respect to the
Brownian filtration Fs , s ≤ t, together with the traduction of the Markov property,
Ns := E(|f˜t0 (z)|p | Fs ) = |f˜s0 (z)|p F̃ (zs , t − s),
(121)
p
zs := f˜s (z)/ ξs ,
(122)
F̃ (z, t) := E(|f˜t0 (z)|p ).
(123)
where
and
Following step by step the argument therein, we write in our new setting
r
κ
iθ
˜
dBs ,
zs = re , d log zs = d log r + idθ = d log fs − i
2
where
z2 + 1
df˜s
= s2
d log f˜s =
ds.
zs − 1
f˜s
Using here a somehow redundant notation in terms of r, θ and ρ := r2 , α := 2θ, we
get
r
zs2 + 1
κ
d log r + idθ = 2
ds − i
dBs ,
zs − 1
2
ρ4 + 8ρ2 − 6ρ3 cos α − 2ρ cos α − 1
,
∂s log |f˜s0 (z)| =
(ρ2 − 2ρ cos α + 1)2
p
2ρ sin α
dθ = − 2
ds − κ/2dBs ,
(ρ − 2ρ cos α + 1)
dr
(ρ2 − 1)
=
ds.
r
(ρ2 − 2ρ cos α + 1)
Writing F̃ (z, t), as defined in (123), as F̃ (r, θ, t), the vanishing of the ds term in the
Itô derivative of Ns gives a PDE in (r, θ, t) satisfied by F̃ , similar to Eq. (96). To
finish, a large t limit argument, entirely similar to (97)-(98) in section 3.2.1, leads
to the following PDE satisfied by F (z) (119), still using at this moment the mixed
notation in r, θ, and ρ := r2 , α := 2θ:
4
(ρ2 − 1)
ρ + 8ρ2 − 6ρ3 cos α − 2ρ cos α − 1
p
+
1
F
+
rFr
(ρ2 − 2ρ cos α + 1)2
ρ2 − 2ρ cos α + 1
2ρ sin α
κ
− 2
Fθ + Fθθ = 0.
(124)
ρ − 2ρ cos α + 1
4
34
Retaining (ρ, α) as the only variables, this finally gives:
4
ρ + 8ρ2 − 6ρ3 cos α − 2ρ cos α − 1
2ρ(ρ2 − 1)
Fρ
p
+
1
F
+
(ρ2 − 2ρ cos α + 1)2
ρ2 − 2ρ cos α + 1
4ρ sin α
Fα + κFαα = 0.
(125)
− 2
ρ − 2ρ cos α + 1
In terms of the z, z variables, writing F = F (z, z̄), this equation becomes
κ
z2 + 1
z2 + 1
− (z∂z − z∂z )2 F + 2
z∂z F + 2
z∂z F
4
z −1
z −1
2
1
1
2
−
+p
+
−
+ 2 F = 0.
1 − z 2 1 − z 2 (1 − z 2 )2 (1 − z 2 )2
(126)
Naturally, defining ζ = z 2 , we can also rewrite this equation in the ζ, ζ variables,
with now F = F (ζ, ζ̄), as
2
ζ +1
κ
ζ +1
ζ∂ζ − ζ∂ζ F +
ζ∂ζ F +
ζ∂ζ F
2
ζ −1
ζ −1
1
2
1
2
p
−
+
+
− 2 F = 0.
− −
2
1 − ζ 1 − ζ (1 − ζ)2 (1 − ζ)2
−
(127)
Remark 3.3. In the ζ, ζ̄ variables, the equation (127) for the oddified moment
function has exactly the same differential part as the BS one (101), the difference
being only in the singular function B(ζ) + B(ζ̄) multiplying the F term. Notice
that this function also vanishes at ζ = ζ̄ = 0, hence fom Lemma 3.1, the space of
solutions which are double power series is one-dimensional.
3.3.2
Special solutions
We can now argue as in section 3.2.2, and look for solutions of (127) of the form
F (ζ, ζ̄) = ϕα (ζ)ϕα (ζ̄)P (ζ ζ̄),
where ϕα (ζ) := (1 − ζ)α ; P is thus rotationally invariant.
The restriction of Eq. (127) to ζ̄ = 0 gives the boundary operator:
κ
ζ +1
p
1
2
2
P(∂)ϕ := − (ζ∂ζ ) ϕ +
ζ∂ζ ϕ +
−
+ 1 ϕ,
2
ζ −1
2 1 − ζ (1 − ζ)2
resulting in the boundary equation P(∂)(ϕα ) = 0. One easily finds
P(∂)(ϕα ) = Aϕα + Bϕα−1 + Cϕα−2 ,
35
(128)
(129)
with
A = −κα2 /2 + α + p/2,
B = κα2 − κα/2 − 3α + p/2,
C = −κα2 /2 + κα/2 + 2α − p.
(130)
(131)
(132)
As before, we see that A + B + C = 0, and solving for A = B = 0 now gives the
special set of values
α=
(8 + κ)(2 + κ)
8+κ
, p = p2 (κ) :=
.
3κ
9κ
(133)
For these values of α, we look for a solution P (ζ ζ̄) of the singular equation
analogous to Eqs. (106) and (107) in section 3.2.2. Because of Remark 3.3, when
plugging the factorized form (128) into Eq. (127), one obtains the same singular
operator as in (103):
ζ +1
κ
ζ̄ + 1 ¯
κ
2
2
¯
ζ̄ ∂ ϕ P.
ζ∂ϕ + ϕ
Psing (D)[P ] = − ϕ̄(ζ∂) ϕ − ϕ(ζ̄ ∂) ϕ + ϕ
2
2
ζ −1
ζ̄ − 1
As a consequence, the singular equation Psing (D)[P ] = 0 is the same as in (106),
with the same solution (108), now in the ζ, ζ̄ variables:
κα2
.
(134)
2
We thus can use Remark 3.3 and Lemma 3.1 to conclude to the unicity of the
solution F to (127) with value 1 at 0. This yields, in the original z variable:
P (ζ ζ̄) = (1 − ζ ζ̄)−β , β =
Theorem 3.6. The oddified whole-plane SLEκ map h0 (z) has derivative moments
E (h00 (z))p/2 = (1 − z 2 )α ,
(1 − z 2 )α (1 − z̄ 2 )α
E |h00 (z)|p =
,
(1 − z 2 z̄ 2 )β
for the special set of exponents p = κα(α + 1)/4 = (8 + κ)(2 + κ)/9κ, with α =
(8 + κ)/3κ and β = κα2 /2 = (8 + κ)2 /18κ.
Notice that p = 2 if and only if κ = 4, in which case α = 1 , β = 2.
Corollary 3.7. For κ = 4, the oddified whole-plane SLEκ map h0 (z) has first and
second derivative moments:
(1 − z 2 )(1 − z̄ 2 )
E(h00 (z)) = 1 − z 2 , F (z, z̄) = E(|h00 (z)|2 ) =
.
(1 − z 2 z̄ 2 )2
By considering those terms which are powers of (z z̄)2 in the double expansion
of F (z, z̄) in Corollary 3.7, one finally proves assertion (iii) in Theorem 3.3 for
the oddified whole-plane SLE, i.e., when the driving function of the whole-plane
Loewner process is the exponential of a Brownian motion.
36
3.3.3
Oddified Lévy-Loewner Evolution
In the case where the driving function in the oddified Loewner equation (117) is the
complex exponential of a Lévy process (Lt ),
λ(2t) = ξt := eiL2t ,
(135)
and in order to compute F (z, z̄) (119), we follow the same martingale argument as
in Section 3.3.1 (see Eqs. (120), (121), (122), and (123)).
1
√ The characteristic function of the exponential’s argument, 2 L2t , associated with
ξt is
i E eξ 2 L2t = e−2tη(ξ/2) .
The Lévy generator Λ is thus defined by its action on the characters
n
Λ(einθ ) := −2η
einθ , n ∈ Z.
2
(136)
Eq. (124) in Section 3.3.1 now becomes
4
(ρ2 − 1)
ρ + 8ρ2 − 6ρ3 cos α − 2ρ cos α − 1
+
1
F
+
rFr
p
(ρ2 − 2ρ cos α + 1)2
ρ2 − 2ρ cos α + 1
2ρ sin α
− 2
Fθ + ΛF = 0.
ρ − 2ρ cos α + 1
By retaining, as in Eq. 125, ρ = r2 and α = 2θ as the only variables, this finally
gives:
4
ρ + 8ρ2 − 6ρ3 cos α − 2ρ cos α − 1
2ρ(ρ2 − 1)
p
+
1
F
+
Fρ
(ρ2 − 2ρ cos α + 1)2
ρ2 − 2ρ cos α + 1
4ρ sin α
Fα + Λ̃F = 0,
− 2
ρ − 2ρ cos α + 1
where the rescaled generator Λ̃ is defined so that
Λ̃(einα ) = −2η(n)einα , n ∈ Z.
(137)
In terms of the z, z variables, and writing F = F (z, z), this equation becomes
z2 + 1
z2 + 1
ΛF + 2
z∂z F + 2
z∂z F
z
−
1
z
−
1
1
1
2
2
+p
+
−
−
+ 2 F = 0,
1 − z 2 1 − z 2 (1 − z 2 )2 (1 − z 2 )2
where the original generator Λ (136) now acts on monomials as
k−l
k `
Λ(z z ) = −2η
zk z`.
2
37
(138)
In the p = 2 case, observe that F (z, z̄) in Corollary 3.7 involves only chiral terms
of the form e±2iθ . For n = ±2, the Lévy generator Λ (136) acts on these terms by
multiplying them by −2η(1). If η1 := η(1) = 2, we see that this action is the same
as that of the Brownian generator in the SLEκ=4 case. Therefore Corollary 3.7 for
the derivative second moment goes through, in the oddified Lévy setting, under the
sole condition that η1 = 2. This is generalized in the following Section.
3.3.4
Oddified Lévy-Loewner moments
Theorem 3.7. If a Lévy process has its first m (≥ 1) symbols given by ηj =
κj 2 /2, 1 ≤p
j ≤ m, with κ = 8/(3m − 1), then the associated odd Lévy-Loewner map
h0 (z) = z f0 (z 2 )/z 2 , where f0 (z) is the whole-plane LLE, has the same derivative moments of order p as for SLEκ , for the particular value of the exponent,
p = 2m(m + 1)/(3m − 1), as given in Theorem 3.6 with α = m, β = 4m2 /(3m − 1).
Remark 3.4. The case m = 1 gives the condition η1 = 2 with an equivalent SLEκ
parameter κ = 4, with p = 2, α = 1, β = 2. Case (iii) of Theorem 3.3 thus follows.
Proof. For the odd case of the whole-plane Lévy-Loewner evolution, the BS-like
equation (138) becomes, when setting F = F (ζ, ζ) in the ζ = z 2 variable,
ζ +1
1
2
ζ +1
1
2
Λ̃F +2
ζ∂ζ F +2
ζ∂ζ F +p
+
−
−
+ 2 F = 0.
ζ −1
1 − ζ 1 − ζ (1 − ζ)2 (1 − ζ)2
ζ −1
(139)
k `
The action of the Lévy infinitesimal generator Λ̃ (137) on a term ζ ζ̄ is
Λ̃ [ζ k ζ̄ ` ] = −2η(k − `)ζ k ζ̄ ` ,
where η(·) here is real and even. It is such that η(0) = 0, therefore for any n ∈ Z,
Λ̃ [(ζ ζ̄)n ] = 0.
For the set of solutions F (ζ, ζ̄) = (1 − ζ ζ̄)−β ϕα (ζ)ϕα (ζ̄), as given in Theorem 3.6,
we thus have
1
Λ̃F (ζ, ζ̄) =
Λ̃[(1 − ζ)α (1 − ζ̄)α ].
(140)
(1 − ζ ζ̄)β
If the exponent α = (8 + κ)/3κ equals an integer m ≥ 1, ϕα = (1−ζ)α is polynomial
of order m, and Λ̃[ϕα ϕ̄α ] contains only the finite set of Lévy symbols {η1 , . . . , ηm }.
If this set coı̈ncides with the set of values (κ/2)`2 for ` = {1, · · · , m}, the action of
the Lévy generator Λ̃ on ϕα ϕ̄α coı̈ncides with that of the Brownian generator. In
this case, F (ζ, ζ̄), solution to the SLEκ equation (127) is also solution to the LévyLoewner differential equation (139), and Theorem 3.6 is also valid for an oddified
Lévy-Loewner evolution with such symbols.
38
3.4
Generalization to processes with m-fold symmetry
The preceding results may be generalized to the case of functions with m-fold
symmetry. These are functions of the form [f (z m )]1/m with f ∈ S and m ∈
N, m ≥ 1. The case of odd functions corresponds to m = 2; equivalently, the
functions withPm−fold symmetry are functions in S whose Taylor series has the
∞
mk+1
form f (z) =
. As for the oddification case, we can associate to
k=0 amk+1 z
(m)
f0 , where (ft ) is a whole-plane SLEκ , its m-folded version h0 (z) := [f0 (z m )]1/m .
By setting ζ = z m , we obtain the following Beliaev-Smirnov-like equation for
(m)
F (ζ, ζ̄) := E |(h0 )0 (z)|p
ζ +1
ζ +1
κ
ζ∂ζ F +
ζ∂ζ F
− (ζ∂ζ − ζ∂ζ )2 F +
2
ζ −1
ζ −1
m
p m−1 m−1
m
+
−
+
−
+ 2 F = 0.
m 1−ζ
(1 − ζ)2 (1 − ζ)2
1−ζ
(141)
We then look for special solutions of the form ϕα (ζ)ϕα (ζ̄)P (ζ ζ̄) where ϕα (ζ) :=
(1 − ζ)α = (1 − z m )α . For ϕα , we look for solutions of the boundary equation for
ζ̄ = 0
κ
ζ +1
p m−1
m
2
P(∂)[ϕα ] = − (ζ∂ζ ) ϕα +
ζ∂ζ ϕα +
−
+ 1 ϕα = 0.
2
ζ −1
m 1−ζ
(1 − ζ)2
We identically have
P(∂)[ϕα ] = Aϕα + Bϕα−1 + Cϕα−2 ,
κ
p
A = − α2 + α + ,
2
m
κ
1
B =
α(2α − 1) − 3α + 1 −
p,
2
m
κ
C = − α(α − 1) + 2α − p.
2
(142)
(143)
(144)
(145)
Notice that we again have the identity A + B + C = 0. Setting the conditions
A = 0, C = 0 so that (142) vanishes, gives the special set of values:
α = αm (κ) :=
2m + 4 + κ
m(2m + 4 + κ)(2 + κ)
, p = pm (κ) :=
.
(m + 1)κ
2(m + 1)2 κ
(146)
For P , Eq. (141) shows that the resulting singular equation (106), Psing (D)[P ] = 0,
does not depend on m, so that P = (1 − ζ ζ̄)−β , with β = κα2 /2.
39
(m)
Theorem 3.8. The m-fold whole-plane SLEκ map h0 (z) has derivative moments
p/2 (m)
E (h0 )0 (z)
= (1 − z m )α ,
(m)
(1 − z m )α (1 − z̄ m )α
E |(h0 )0 (z)|p =
,
(1 − z m z̄ m )β
for the special set of exponents p = pm (κ) = m(2m + 4 + κ)(2 + κ)/2(m + 1)2 κ,
with α = (2m + 4 + κ)/(m + 1)κ and β = κα2 /2 = (2m + 4 + κ)2 /2(m + 1)2 κ.
The case p = 2 is of special interest, since it allows one to find the moments
E (|amk+1 |2 ) from Plancherel formula. Setting p = 2 in the above, and solving for κ
yields
2(m + 2)
.
(147)
κ = 2m, or κ =
m
In the first case, we have α = 2/m, β = 4/m, thus
F (ζ, ζ̄) =
(1 − ζ)(1 − ζ̄)
(1 − ζ ζ̄)2
m+2
,
m
In the second case, we have α = 1, β =
F (ζ, ζ̄) =
(1 − ζ)(1 − ζ̄)
(1 − ζ ζ̄)
m+2
m
2/m
; κ = 2m.
(148)
2(m + 2)
.
m
(149)
and
; κ=
Let us detail all possibilities with m ≤ 4:
(i) m = 1 yields the two cases κ = 6, κ = 2, corresponding to the two first cases
of theorem 1.2.
(ii) m = 2 gives rise to the single value κ = 4, and to the third case in theorem
1.2.
(iii) m = 3 corresponds to κ = 6 and κ = 10/3, with respective F -functions
F (ζ, ζ̄) =
(1 − ζ)2/3 (1 − ζ̄)2/3
(1 − ζ)(1 − ζ̄)
; F (ζ, ζ̄) =
.
4/3
(1 − ζ ζ̄)
(1 − ζ ζ̄)5/3
(iv) For m = 4, one gets κ = 8 or 3, with respective F -functions:
F (ζ, ζ̄) =
(1 − ζ)1/2 (1 − ζ̄)1/2
(1 − ζ)(1 − ζ̄)
; F (ζ, ζ̄) =
.
1 − ζ ζ̄
(1 − ζ ζ̄)3/2
40
Let us return for p = 2 to general values of m, with κ given by (147), and compute
E (|amk+1 |2 ). Write
∞
X
α
(1 − x) =
λk (α)xk ,
k=0
with coefficients λk (α) := ((−1)k /k!) α(α − 1) · · · (α − k + 1). In the first case,
κ = 2m, we have from (148)
Pk
2
E |amk+1 |
In the second case, κ =
2(m+2)
,
m
=
j=0
λ2j (2/m)|λk−j (−4/m)|
.
(mk + 1)2
(150)
Eq. (149) gives
Qk−1
m+2
m+2
λ
(
)
+
λ
(
)
k
k−1
j=0 (jm + 2)
m
m
=
;
E |amk+1 |2 =
2
(mk + 1)
(mk + 1)mk k!
(151)
for m = 1, 2 one recovers the values already computed.
In conclusion, we have found infinitely many cases where one may exactly compute the variances of the coefficients of whole plane SLE. The following cases correspond to some physically relevant situations:
(a) m = 1, κ = 6 with formula (151) (percolation [63, 65]);
(b) m = 1, κ = 2 with formula (150) (loop-erased random walk [43, 63]);
(c) m = 2, κ = 4 with formula (150) or (151) (Gaussian free field contour lines [64]);
(d) m = 3, κ = 6 with formula (150) (percolation [65]);
(e) m = 4, κ = 8 with formula (150) (spanning trees [63]);
(f) m = 4, κ = 3 with formula (151) (critical Ising model [9, 66]);
(g) m = 6, κ = 8/3 with formula (151) (self-avoiding walk [19, 45]).
41
4
Multifractal spectra for infinite whole-plane SLE
The aim of this section is to establish Theorem 1.3: that the averaged integral means
spectra, as defined in (8), for the interior whole-plane SLE map f0 (z); its associated
(m)
odd map h0 (z); or its m-fold transform h0 (z), have a phase transition for p large
enough (resp. above (18), (19), (20)), where they are bounded below by the right
hand-side of formulae (15), (16), (17), respectively. We believe these bounds to be
exact.
Let us begin with some relevant results for integral means spectra and related
multifractal spectra.
4.1
4.1.1
Integral means spectrum
SLE’s harmonic measure spectra
The general theory of the integral means spectra, and of the associated multifractal
properties of the harmonic measure, have been the subject of important pioneering
works, among which stand out those of L. Carleson, P. Jones and N. Makarov
[8, 37, 51, 52]. The search for universal spectra, which provide universal functions
as upper-bounds, have lead to well-known results and conjectures which we briefly
recall below (Section 4.1.2).
In the case of conformally invariant critical curves, i.e., SLEs, the multifractal
spectrum associated with the harmonic measure near those curves was first obtained from quantum gravity methods by the first author [16, 15, 17, 18, 19, 20].
This was extended to the mixed multifractal spectrum describing both the singularities and the winding of equipotentials near a conformally invariant curve [21].
Another heuristic derivation of the harmonic measure spectrum was obtained from
a Laplacian growth equation, similar to Eq. (99) here, but for chordal SLE [33].
The corresponding SLE integral means spectrum was later rigorously established,
in an expectation sense, by Beliaev and Smirnov [4], starting from Beliaev’s thesis
[3]. These authors used the very same equation as Eq. (99) here, that they derived
precisely for that purpose, in their case for the exterior whole-plane SLE.
Another method, the so-called “Coulomb gas” approach of conformal field theory,
is also applicable [6, 61], and was extended to the mixed multifractal spectrum [5, 22].
Amazingly, these predictions for the fine structure of the harmonic measure were
tested numerically, and successfully, for percolation and Ising clusters [1].
Let us mention that Chen and Rohde obtained in Ref. [10] derivative estimates
for the (chordal) Schramm-Loewner evolution driven by a symmetric α-stable process, and showed that its hull has Hausdorff dimension 1, thereby presenting a
non-multifractal behavior. Similar results was found by Johansson and Sola [35] for
a random growth model obtained by driving the Loewner equation by a compound
Poisson process. We therefore restrict our study here to the interior whole-plane
42
SLE curve.
In the radial setting, the integral means spectrum (7)-(8) is associated with the
divergent behavior of the moments of order p of the map derivative’s modulus |f 0 (z)|
near the unit circle ∂D, possibly augmented by the extra singular behavior of the
map at point z = 1, where the SLE driving function originates at time t = 0. When
the latter singular behavior starts to dominate, in fact when p becomes negative
enough [4], the integral means spectrum undergoes a phase transition, after which
the harmonic measure behavior is dominated by the tip of the SLE curve. This was
observed in Ref. [33], while the tip spectrum was later obtained rigorously, in the
sense of expectations in Ref. [4], and in an almost sure sense in Ref. [36].
4.1.2
Universal spectra
Given f holomorphic and injective in the unit disk, we define, for p ∈ R,
R 2π
ln 0 |f 0 (reiθ )|p dθ
,
βf (p) = lim sup
1
ln 1−r
r→1−
so that βf (p) is the smallest number q such that there exists a C > 0
Z 2π
C
|f 0 (reiθ )|p dθ ≤
(1 − r)q+ε
0
as r → 1, and for every ε > 0.
Theorem 4.1. If f is holomorphic and injective in the unit disk, then
βf (p) ≤ 3p − 1, 2/5 ≤ p < ∞.
If moreover f is bounded, then
βf (p) ≤ p − 1, p ≥ 2.
Both exponents are sharp, the first one being attained for the Koebe function.
This theorem is due to Feng and McGregor [29]. For a proof, consult, e.g., [57].
We will need below the following variant of this theorem:
Theorem 4.2. Let f be a function injective and holomorphic in the unit disk, such
that f (0) = 0, and let us denote by h(z) its oddification:
p
h(z) := z f (z 2 )/z 2 .
Then we have βh (p) ≤ 2p−1 for p ≥ 2/3. This bound is attained for the oddification
of the Koebe function.
43
This variant is originally due to Makarov [52]; a proof that parallels that of Feng
and McGregor is given below in Appendix C 5.3.
In the sequel we will need some facts about three different universal spectra,
respectively for the schlicht class S, the subclass Sb of bounded functions, and the
subclass So of odd functions. For p ∈ R, define:
BS (p) := sup{βf (p), f ∈ S}; BSb (p) := sup{βf (p), f ∈ Sb };
BSo (p) := sup{βf (p), f ∈ So }.
The BSb spectrum is the most studied one in the litterature. By Remark 4.1 and
Theorem 4.2, respectively:
BS (p) = 3p − 1 for p ≥ 2/5; BSb (p) = p − 1 for p ≥ 2;
BSo (p) = 2p − 1 for p ≥ 2/3.
For the sake of completeness, let us briefly recall some known or conjectured results
about these universal spectra. The main one is Brennan conjecture, which reads:
B(−2) = 1. If true, this conjecture would imply that BSb (p) = |p| − 1, for p ≤ −2.
This is not known, but Carleson and Makarov [8] have shown that there exists
p0 ≤ −2 such that BSb (p) = |p| − 1 for p ≤ p0 . Notice that BSb (p) = |p| − 1 for
p ≥ 2. A rather speculative conjecture, named after Kraetzer, asserts that
BSb (p) =
p2
, −2 ≤ p ≤ 2.
4
Makarov [52] has proven that
BS (p) = max(BSb (p), 3p − 1),
so that if both Kraetzer and Brennan conjectures are true, then for |p| ≤ −2, BS (p) =
√
√
2
|p| − 1, for −2 ≤ p ≤ 6 − 4 2, BS (p) = p4 , and for p ≥ 6 − 4 2, BS (p) = 3p − 1.
In our study, the unbounded character of the whole-plane maps under consideration plays a crucial role for the spectrum. This can already be seen in the
limit κ → 0, where the spectra should converge to that of the Koebe function (1),
K(z) = z/(1 + z)2 , hence to βK (p) = 3p − 1, or to βKo (p) = 2p − 1 for its oddified
version Ko (z) = z/(1 + z 2 )2 .
The theorem 4.2 can be generalized to the class Sm defined for nonzero m ∈ N
as the set of functions of the form
h(m) (z) := z[f (z m )/z m ]1/m , f ∈ S.
The case of odd functions above corresponds to m = 2. We may define as above
BSm (p) := sup{βh (p) , h ∈ Sm },
and a straightforward adaptation of the proof of Theorem 4.2 (as given in Appendix
C 5.3) gives [52] the
44
Theorem 4.3. For p ≥
2m
m+4
we have
BSm (p) =
4.2
4.2.1
m+2
p − 1.
m
Integral means spectrum for infinite whole-plane SLE
Restriction to the unit circle
The method introduced in [4] consists in finding approximate solutions to equation
(99) in the vicinity of the unit circle for |z| > 1, and near z = 1. We adapt it here
to the inner whole-plane equation (99), with σ = −1 and |z| < 1,
4
r(r2 − 1)
r + 4r2 (1 − r cos θ) − 1
−
σ
Fr
F
+
p
(r2 − 2r cos θ + 1)2
r2 − 2r cos θ + 1
2r sin θ
κ
− 2
Fθ + Fθθ = 0.
(152)
r − 2r cos θ + 1
2
It is convenient to write this equation as
r(r2 − 1)
2r sin θ
κ
N (r, θ)
−
σ
F
+
Fr −
Fθ + Fθθ = 0,
p
2
D (r, θ)
D(r, θ)
D(r, θ)
2
(153)
with
D(r, θ) := r2 − 2r cos θ + 1 = |1 − z|2 ,
N (r, θ) := r4 + 4r2 (1 − r cos θ) − 1
= 2r2 D(r, θ) + (r − 1)(r3 − r2 + 3r + 1).
(154)
(155)
We then look for approximate (but possibly exact) solutions of the form
ψ(r, θ) = (1 − r2 )−β g(r2 − 2r cos θ + 1) = (1 − z z̄)−β g(|1 − z|2 ).
(156)
Let us first remark that for any given value of κ, and for the special values p = p(κ)
and α = α(κ) given in (114), the exact solution found in Theorem 3.4 is precisely of
the form (156), with β = (κ/2)α2 and g(x) = xα . It is thus necessarily a solution,
for p = p(κ), to the following explicit equation, obtained from (153), (1 − r2 )−β
being further factored out:
N (r, θ)
2r2
r(1 − r2 )
p
−
σ
g
−
β
g
−
−
(2r − 2 cos θ)g 0
D2 (r, θ)
D(r, θ)
D(r, θ)
4r2 sin2 θ 0 κ
−
g + 2r cos θ g 0 + 4r2 sin2 θ g 00 = 0,
(157)
D(r, θ)
2
where g = g(r2 − 2r cos θ + 1).
45
When p is not equal to the special value p(κ) of Eq. (114), the trial exponent
β and the function g are determined from the restriction of Eq. (157) to the unit
circle ∂D. One observes that on the unit circle
D(r = 1, θ) = 2 − 2 cos θ; N (1, θ) = 2D(1, θ); g = g(2 − 2 cos θ).
(158)
Setting r = 1 in Eq. (157) and factoring out [D(1, θ)]−1 , we therefore arrive at
κ
p [2 − σD(1, θ)] g − 2β g − 4 sin2 θ g 0 + D(1, θ) 2 cos θ g 0 + 4 sin2 θ g 00 = 0. (159)
2
Define now x := 2 − 2 cos θ = D(1, θ), such that 0 ≤ x ≤ 4; the equation on the
unit circle simply becomes
hκ
i
κ
[p(2 − σx) − 2β] g + (2 − x) − (4 − x) x g 0 + (4 − x)x2 g 00 = 0.
(160)
2
2
By homogeneity, for a function of the power law form g(x) = xγ , the left-hand side
of (160) becomes (a + bx)g(x), with
a = 2p − 2β + (κ − 4)γ + 2κγ(γ − 1),
κ
κ
γ − γ(γ − 1).
b = −σp + 1 −
2
2
We thus get a power law solution to (160) if and only if a = 0 and b = 0, i.e.,
γ
β = p − (κ + 4) + κγ 2 ,
2
2κ
γ − γ + σp = 0.
2
(161)
(162)
Recall that for the inner whole-plane SLE considered here, we have σ = −1, while
in the BS case [4] one had σ = +1. This yields here
γ
β = p − (κ + 4) + κγ 2 ,
2
2κ
γ − γ − p = 0.
2
(163)
(164)
We thus obtain the pair of power law solutions ψ± := (1 − z z̄)−β± g(|1 − z|2 ), g(x) =
xγ± with
p
κ
1
(165)
β± (p) = 3p − γ± = 3p − 1 ± 1 + 2κp ,
2 p
2
γ± (p) = κ−1 1 ± 1 + 2κp .
(166)
Besides the power law solutions xγ obtained here, the second order differential equation (160) has another type of independent solution, which will be studied in section
4.2.5.
46
4.2.2
Action of the differential operator
We thus arrive at the function
ψ(r, θ) = (1 − r2 )−β g(r2 − 2r cos θ + 1) = (1 − r2 )−β (r2 − 2r cos θ + 1)γ ,
(167)
with β = β± and γ = γ± , as given in Eqs. (165) and (166). It satisfies, asymptotically close to the unit circle, the same equation (152) as F (r, θ) = F (z, z̄) =
E(|f00 (z)|p ). This function is of the form ψ(r, θ) = ψ(z, z̄) = (1 − z z̄)−β xγ , where
now
x := r2 − 2r cos θ + 1 = |1 − z|2 = (1 − z)(1 − z̄) = 1 − (z + z̄) + z z̄,
(168)
hence
ψ(z, z̄) = (1 − z z̄)−β ϕγ (z)ϕγ (z̄) = (1 − z z̄)−β |1 − z|2γ .
(169)
It will prove useful to evaluate the action of the differential operator P(D) on (169)
for general values of β and γ by using (103), (106) and (110). The general result,
using A + B + C = 0, is
P(D)ψ(z, z̄) =
ψ(z, z̄)x−1
(170)
1 − z z̄
2
+1 −2 ,
× (κγ − 2β)z z̄ − A(1 − z z̄ − x) + C (1 − z z̄)
x
where A is given by (111) and C by (113). At this stage, it is important to make
the following remark about the comparison of the standard BS spectrum (253) to
the formulation here:
Remark 4.1. • A study of the singularity analysis carried out in Ref. [4] shows
that the standard BS spectrum β0 (p) = κγ02 /2 (253) (see Eqs. (11) and (12) there)
is obtained in our notation precisely from the equation C = κγ02 /2 − β0 = 0, with a
parameter γ0 determined from C(γ0 ) = 0, where C(γ) := − κ2 γ(γ − 1) + 2γ − p as
in Eq. (113). The corresponding approximate solution to the differential equation
(157) near the unit circle, as obtained in [4] for σ = +1, can thus be extended to
our inner whole-plane case with σ = −1 since C is independent of σ.
• Alternatively, in the function considered here, the equation (164) that determines
the different parameter γ (166) coı̈ncides with A(γ) := − κ2 γ 2 + γ + p = 0 in (111),
while (163) for β and (113) for C together still give C = κγ 2 /2 − β.
• Therefore, both the BS solution and the ψ function studied here share C = κγ 2 /2−
β. For BS, C = 0, whereas here A = 0.
If we thus set C = κα2 /2 − β into the equation (170) above, we get
1 − z z̄
−1
2
P(D)ψ(z, z̄) = ψ(z, z̄)x
(κγ /2 − β)
− A (1 − z z̄ − x).
x
47
(171)
For the BS solution [4], the first term in the bracket (the most singular term for
x → 0, i.e., z → 1) vanishes, while in our case, the second term A vanishes. For BS,
one thus has
p
1 4 + κ ± (4 + κ)2 − 8κp ,
(172)
γ0± (p) :=
2κ
1 ± 2
β0± (p) :=
κγ (p) ,
(173)
2 0
1
β0 (p) := β0− (p) = κγ02 , γ0 := γ0− ,
(174)
2
where the lower branch γ0 := γ0− is the one selected among the two solutions γ0± to
C = 0 (see Eq. (11) in [4] and Eqs. (9), (10) and (253) here).
Recall that the special point (114) (α(κ) > 0, p(κ)) of theorem 3.4 was obtained
as the intersection of conditions (111) A = 0 and (113) C = 0, together with
β = κα2 /2. This leads to the following remark.
Remark 4.2. The special point (α(κ), p(κ)) is at the intersection of the curve γ+ (p)
(166) and of the complementary BS curve γ0+ (p) (172). We thus have at the special
point
β = β+ = β0+ = κα2 /2, α = γ+ = γ0+ .
(175)
When checking the conditions necessary for finding a positive bounded BS solution with the proper regularity properties at z = −1, as done in [4], one finds the
following
Proposition 4.1. For σ = +1, the BS solution (172), (174) for the average integral
means spectrum, while analytic up to p = (4 + κ)2 /8κ, exists only up to p0 (κ) :=
3(4 + κ)2 /32κ, after which it stays linear [4]. For σ = −1, the BS solution only
exists for p ≤ p∗ (κ), where p∗ (κ) is given by (18):
p
1 2
2
(4 + κ) − 4 − 2 4 + 2(4 + κ) .
p (κ) =
16κ
∗
(176)
Note that ∀κ ≥ 0, p∗ (κ) < p0 (κ).
Proof. Lemma 5 in [4] introduces the quantities (for σ = +1 and denoted there by
a and b)
1 1p
−
1 − 2σκp
κ κ
1 1p
:= γ0 − +
1 − 2σκp
κ κ
a0 := γ0 −
(177)
b0
(178)
where γ0 = γ0− is the lower solution to C = 0 in γ0± (172).
48
A first condition for the existence of a bounded BS solution when z → −1 on the
circle ∂D is the non-negativity condition 1/2 − a0 − b0 ≥ 0, which gives p ≤ 3(4 +
κ)2 /32κ independently of σ. Then a second condition is Γ(1/2 − a0 )Γ(1/2 − b0 ) > 0.
For σ = +1, this is fulfilled for p ≥ 1/2κ by complex conjugation, whereas for
p ≤ 1/2κ BS show that 1/2 − a0 ≥ 1/2 − b0 > 0.
For σ = −1, the situation turns out to be different. One has indeed from (166)
1 1 1p
1
− b0 =
+ −
1 + 2κp − γ0
2
2 κ κ
1 2
+ − γ+ − γ0 .
=
2 κ
(179)
Since γ+ and γ0 are both increasing functions of p there may be a point where
1/2 − b0 = 0, after which it becomes negative and the BS solution for σ = −1 ceases
to exist. Let us now recall that the identity
β = κγ 2 /2 − C(γ) = κγ 2 − (2 + κ/2)γ + p
(180)
is satisfied by the four pairs (β0 ± , γ0 ± ) and (β± , γ± ). Notice that the point p∗ (κ) (18)
is defined by β0 (p) = β0− (p) = β+ (p). Thus the corresponding γ0 = γ0− and γ+ are
the two solutions of the quadratic equation (180), such that γ0 + γ+ = (2+ κ/2)/κ =
2/κ + 1/2, which is precisely the point where 1/2 − b0 (179) vanishes.
4.2.3
Beyond the BS solution: p ≥ p∗ (κ).
In that range, we now look at the properties of the function ψ(z, z̄) (169), where β
and γ satisfy the relation (163) and A(γ) = 0 (164), and are given by the pair of
solutions (165) and (166). Eq. (171) then yields the explicit result:
κ
2
−β+1
γ − β (2 − z − z̄ − 2x)
P(D)[ψ(z, z̄)] = (1 − z z̄)
ϕγ−2 (z)ϕγ−2 (z̄)
κ
2
= (1 − z z̄)−β+1 xγ−2
γ 2 − β (1 − z z̄ − x)
(181)
2
κ
= ψ(z, z̄)
γ 2 − β (1 − z z̄)(1 − z z̄ − x)x−2 .
(182)
2
The quantity in factor of ψ(z, z̄) in (182) vanishes both on the unit circle ∂D and
on the circle ∂D1/2 := {z : 1 − z z̄ − x = 0}, centered at (1/2, 0) and of radius 1/2,
which passes through z = 0 and z = 1, and is tangent to the unit circle ∂D at z = 1
(see Fig. 4). The overall sign of (182) crucially depends on the position of z with
respect to the circle ∂D1/2 : For z inside the disk
D1/2 := {z : 1 − z z̄ − x > 0},
(183)
(182) has the same sign as the coefficient κγ 2 /2 − β, and the opposite sign when
z lies outside of that disk. The sign of the coefficient κγ 2 /2 − β itself depends on
49
Figure 4: The unit disk D and the disk D1/2 (183). The signs indicated are those
(187) of P(D)[ψ+ ] for p ≤ p(κ), which vanishes on ∂D and ∂D1/2 .
which branch is chosen in (165) and (166). One easily finds that
p
1 1 κ 2
γ − β± =
+
1 ± 1 + 2κp − 2p.
2 ±
κ 2
For the negative branch, and for p ≥ 0, it is clear that
κγ−2 /2 − β− ≤ 0.
(184)
The positive branch κ2 γ+2 − β+ , on the other hand, has a zero for p = p(κ) =
(6+κ)(2+κ)/8κ with γ+ = α = (6+κ)/2κ, which naturally corresponds to the special
set of values (114) where there exits the exact solution (109) with β+ = β = κα2 /2.
One therefore has
κγ+2 /2 − β+ > 0,
κγ+2 /2 − β+ = 0,
κγ+2 /2 − β+ < 0,
p < p(κ),
p = p(κ),
p > p(κ).
(185)
We therefore arrive at the various domain inequalities for p 6= p(κ), with the obvious
notation ψ± (z, z̄) := (1 − z z̄)−β± xγ± ,
P(D)[ψ− (z, z̄)] < 0, z ∈ D1/2 , P(D)[ψ− ] > 0, z ∈ D \ D1/2 ; (186)
p ≤ p(κ) : P(D)[ψ+ (z, z̄)] > 0, z ∈ D1/2 , P(D)[ψ+ ] < 0, z ∈ D \ D1/2 ; (187)
p ≥ p(κ) : P(D)[ψ+ (z, z̄)] < 0, z ∈ D1/2 , P(D)[ψ+ ] > 0, z ∈ D \ D1/2 ; (188)
the resulting ratio P(D)[ψ± ]/ψ± vanishes at the boundaries of the above domains,
i.e., on ∂D and ∂D1/2 . For p = p(κ), ψ+ is an exact solution such that P(D)[ψ+ ] = 0.
50
Logarithmic modification. Following [4], let us consider now the action of the
differential operator on the modified function ψ(z, z̄)`δ (z z̄), where here ψ := ψ±
and where the factor
`δ (z z̄) := [− log(1 − z z̄)]δ , δ ∈ R,
(189)
brings in a (soft) logarithmic singularity. From Eq. (101) one finds the simple result
P(D)[ψ(z, z̄)`δ (z z̄)] = `δ (z z̄)P(D)[ψ(z, z̄)] − ψ(z, z̄)2z z̄(1 − z z̄)x−1 `δ0 (z z̄)
2δz z̄x−1
,(190)
= `δ (z z̄) P(D)[ψ(z, z̄)] − ψ(z, z̄)
[− log(1 − z z̄)]
where the derivative `δ0 (z z̄) is taken with respect to z z̄. Using Eq. (182) yields (here
γ := γ± and β := β± ):
P(D)[ψ(z, z̄)`δ (z z̄)]
= `δ (z z̄)ψ(z, z̄)x−1
(191)
2z z̄δ
κ 2
−1
γ − β (1 − z z̄)(1 − z z̄ − x)x −
×
.
2
[− log(1 − z z̄)]
• Consider first the domain D1/2 (Fig. 4). The sign of P(D)[ψ± (z, z̄)] for z ∈ D1/2 is
given in the three different cases by the first column of Eqs. (186), (187) and (188).
In each case, this sign is also that of the first term of (191) in the same domain and
for the same case. For each case, choose the sign of δ so that the second term in
(191) has the same uniform sign as the first term in D1/2 . Then P(D)[ψ± (z, z̄)] and
P(D)[ψ± (z, z̄)`δ (z z̄)] have the same sign in D1/2 .
• Consider now the complementary domain D \ D1/2 . Take z ∈ D \ D1/2 on the
circle ∂D(r) of radius r < 1 centered at the origin (with ∂D(1) = ∂D), so that
1 − z z̄ = 1 − r2 . The quantity (1 − z z̄ − x)x−1 in (191) is negative in D \ D1/2 and
equals (1 − r2 )x−1 − 1 on ∂D(r), for which −1 < (1 − r2 )x−1 − 1 ≤ 0. On the circle
∂D(r) of radius r < 1 and outside of D1/2 , one thus has the following bounds for
the first term of (191):
−(1 − r2 ) < (1 − z z̄)(1 − z z̄ − x)x−1 ≤ 0.
Therefore, for r close enough to 1, say r1 < r < 1, the second term in (191), which
vanishes logarithmically when r → 1− , dominates the first term, which is of order
O(1−r2 ), hence determines the overall sign of P(D)[ψ± (z, z̄)`δ (z z̄)] for z ∈ D\D1/2 .
• Recall now that δ has been chosen precisely such that the sign of the second
term in (191) is that of P(D)[ψ± `δ ] and P(D)[ψ± ] in D1/2 . We thus conclude that
in the whole annulus r1 < r < 1, the sign of P(D)[ψ± (z, z̄)`δ (z z̄)] is uniform and
given by that of P(D)[ψ± (z, z̄)] for z ∈ D1/2 , as given for the three canonical cases
51
[ψ− ], [ψ+ , p ≤ p(κ)], [ψ+ , p ≥ p(κ)] by the first column in Eqs. (186), (187) and (188).
We therefore conclude that there exist for these three cases (denoted by i =
1, 2, 3) three open annuli A(ri ) := {z : ri < |z| < 1} = D \ D(ri ) whose boundary
includes ∂D, where one has respectively (for a specific sign of δ chosen in each case
as described above):
• P(D)[ψ− (z, z̄)`δ (z z̄)] < 0, z ∈ A(r1 ), so that ψ− (z, z̄)`δ (z z̄) is locally a subsolution
to the equation P(D)[F (z, z̄)] = 0;
• for p < p(κ), ψ+ (z, z̄)`δ (z z̄) is a supersolution with P(D)[ψ+ (z, z̄)`δ (z z̄)] > 0 for
z ∈ A(r2 );
• for p > p(κ), ψ+ (z, z̄)`δ (z z̄) is a subsolution with P(D)[ψ+ (z, z̄)`δ (z z̄)] < 0 for
z ∈ A(r3 );
• for p = p(κ), P(D)[ψ+ (z, z̄)] = 0, z ∈ D, so that ψ+ (z, z̄) = F (z, z̄) = (1 −
z z̄)−β+ |1 − z|2γ+ is the exact solution in Theorem 3.4 with parameters (114): γ+ =
α(κ) and β+ = κα2 /2.
We then follow the same method as in Refs. [3, 4]. The operator P(D),
when written in polar coordinates as in (152), is parabolic, where θ corresponds
to the spatial variable, and r to the time variable [27]. In the above, the functions
ψ± (z, z̄)`δ (z z̄) are positive
functions bounded on the respective circles of radius ri ,
as F (z, z̄) = E f00 (z)|p is. One can thus find positive constants ci such that
F < c1 ψ− `δ , r = r1 ; c2 ψ+ `δ < F, r = r2 , p < p(κ); F < c3 ψ+ `δ , r = r3 , p > p(κ).
Using then in each of the corresponding annuli where P(D)[ψ± `δ ] has a definite sign,
respectively, the maximum principle, the minimum principle, and the maximum
principle ([27], Th. 7.1.9), yields the
Proposition 4.2.
F < c1 ψ− `δ , z ∈ A(r1 ), ∀p,
c2 ψ+ `δ < F, z ∈ A(r2 ), p < p(κ),
F < c3 ψ+ `δ , z ∈ A(r3 ), p > p(κ).
(192)
(193)
(194)
These inequalities will be used in Section 4.2.4 to establish the existence at p∗ (κ)
of a phase transition in the integral means spectrum of the inner whole-plane SLE
and to prove Theorem 1.3.
4.2.4
Proof of Theorem 1.3
Proof. The average integral means spectrum of the whole-plane SLE is given by the
asymptotic behavior for r → 1− of the F integral:
Z 2π
Z 2π
(r→1− )
F (r, θ)dθ =
E(|f00 (r, θ)|p )dθ (1 − r)−β(p) .
(195)
0
0
52
For the ψ function defined in Eqs. (167)-(169), the integral means are:
Z
2π
2 −β
Z
2π
ψ(r, θ)dθ = (1 − r )
g(r, θ)dθ,
(196)
0
0
where we write g(r, θ) = g[D(r, θ)] = Dγ (r, θ) = |1 − z|2γ . This function g has a
singularity at z = 1, i.e., for r = 1, θ = 0. Near that point, its argument D(r, θ) =
r2 − 2r cos θ + 1 is equivalent to D(r, θ) ∼ (1 − r)2 + θ2 .
The exponents β and γ in the above
R 2π are given by Eqs. (165) and (166). For the
(+) branch, γ+ > 0 and the integral 0 g+ (1, θ)dθ is integrable at θ = 0, which is a
zero of g. For the other branch, γ− is negative for p ≥ 0, and the singularity along
the unit circle at θ = 0 is no longer integrable when 2γ− + 1 ≤ 0. This corresponds
to a cross-over value p = p̃(κ) := (4 + κ)/8. One therefore has:
2π
Z
ψ+ (r, θ)dθ
(r→1− )
(1 − r)−β+ ,
(197)
0
2π
Z
ψ− (r, θ)dθ
(r→1− )
(1 − r)−β− , p ≤
0
(r→1− )
4+κ
,
8
(1 − r)−β− +2γ− +1 , p ≥
(198)
4+κ
.
8
(199)
Consider now the modified functions ψ± `δ , where `δ is the weakly diverging or
vanishing logarithmic function (189). The asymptotic power law behaviors of their
integral means near the unit circle are obviously the same as for ψ± :
Z
2π
ψ+ (r, θ)`δ (r2 )dθ
(r→1− )
(1 − r)−β+ ,
(200)
0
Z
2π
ψ− (r, θ)`δ (r2 )dθ
(r→1− )
(1 − r)−β− , p ≤
0
(r→1− )
4+κ
,
8
(1 − r)−β− +2γ− +1 , p ≥
(201)
4+κ
.
8
(202)
By plugging the asymptotic behaviors (195) and (200) into inequalities (193) and
(194) of Proposition 4.2, we obtain
β+ (p) ≤ β(p), p < p(κ) =
(6 + κ)(2 + κ)
,
8κ
β(p) = β+ (p), p = p(κ),
β(p) ≤ β+ (p), p > p(κ) =
(204)
(6 + κ)(2 + κ)
.
8κ
This ends the proof of Theorem 1.3 for the m = 1 case.
53
(203)
(205)
By plugging the asymptotic behaviors (195) and (201), (202) into inequality
(192) of Proposition 4.2, we obtain
4+κ
,
8
4+κ
β(p) ≤ β− (p) − 2γ− (p) − 1, p ≥
.
8
β(p) ≤ β− (p), p ≤
(206)
(207)
At this point, we can invoke the bound implied by the existence of an universal
spectrum BS for the S schlicht class of univalent functions f in the unit disk. As
we have seen in section 4.1.2, the integral means spectrum βf (p) of such an f ∈ S
is bounded, for p ≥ 2/5, as:
βf (p) ≤ BS (p) = 3p − 1, p ≥ 2/5,
the maximum being attained for the Koebe function (1). For p 0, we have from the
definitions (165) and (166)
4+κ
,
8
4+κ
β+ < 3p − 1 < β− ≤ β− − 2γ− − 1, p ≥
.
8
β+ < 3p − 1 < β− , p ≤
(208)
(209)
This therefore excludes the (−) branch as a possible integral means spectrum.
This strongly suggests that the average integral means spectrum of the unbounded inner whole-plane SLEκ is simply given, for p ≥ 0, by
1 1p
1 + 2κp ,
(210)
β(p, κ) = max β0 (p, κ), 3p − −
2 2
where β0 (p, κ) is given by Eq. (10), or Eqs. (253) and (254) if one extends the range
of moment orders to p ≤ 0. The phase transition in Eq. (210) occurs at the critical
point p∗ (κ), as given by Eq. (18), where the BS solution ceases to exist (see Prop.
(4.1)). These conclusions are illustrated in Fig. 5 below.
54
Figure 5: Standard BS branch of the SLEκ=6 bulk average means spectrum β0 (p, κ) =
β0− (p, κ) in Eqs. (10) and (174) (in blue); second “non-physical” BS branch
β0+ (p, κ) in Eq. √(173) (in red); whole-plane multifractal function β+ (p, κ) =
3p − 1/2 − (1/2) 1 + 2κp in Eq. 165 (in green). The resulting average integral
means spectrum β(p, κ) (210) of whole-plane SLEκ undergoes a phase transition at
p = p∗ (κ) (18), where the first two curves intersect, β0 (p∗ , κ) = β+ (p∗ , κ), such that
β(p, κ) = β0 (p, κ), ∀p ∈ [0, p∗ (κ)], and β(p, κ) = β+ (p, κ), ∀p ∈ [p∗ (κ), ∞). Note
that the special point p = p(κ) (114) (here p(6) = 2) lies at the intersection of the
curve β+ (p, κ) with the non-physical BS branch β0+ (p, κ), whereas the phase transition point p∗ (κ) lies at the intersection of the curve β+ (p, κ) with the standard
(“physical”) BS branch β0− (p, κ).
55
4.2.5
General solution
Let us now look for the general solution to (160), by setting g(x) = xγ g0 (x), where
the exponent γ is, as before, determined by (164). Actually, the singularity analysis
in section 4.2.4 indicates that only the (+) branch γ+ , β+ in (166),
p
1
1 + 1 + 2κp ,
(211)
γ = γ+ =
κ
p
κ
1
β = β+ = 3p − γ+ = 3p − 1 + 1 + 2κp
(212)
2
2
will be relevant to the integral means spectrum. Substituting g above into (160)
yields
g000
g00
1 1
1
−a ,
24−x
x
1
2
a :=
+ 2γ − ,
2
κ
=
(213)
(214)
which is readily integrated into
g00 (x) = −C1 (4 − x)−1/2 x−a ,
Z 4
g0 (x) = C0 + C1
(4 − u)−1/2 u−a du,
(215)
(216)
x
with C0 and C1 arbitrary constants. The general solution to (160) is thus
Z 4
γ
γ
g(x) = C0 x + C1 x
(4 − u)−1/2 u−a du.
(217)
x
The one-dimensional space of solutions for which C0 > 0, C1 = 0, is precisely the
one studied in sections 4.2.2 and 4.2.3 above.
Consider now the second set of solutions such that C0 = 0, C1 > 0 (taking C1 = 1
hereafter):
Z 4
g0 (x) :=
(4 − u)−1/2 u−a du; g00 (x) = −(4 − x)−1/2 x−a .
(218)
x
The parameter a is, thanks to (211) (214),
a=
1 2p
+
1 + 2κp.
2 κ
(219)
Recall that actually here we only consider the range p > p∗ (κ) (18), where
the BS
2
solution no longer exists. Note that one has a > 1 for p > (κ /16) − 1 /2κ, while
p∗ (κ) is always greater than the latter. Hence for
p > p∗ (κ), a > 1.
56
The integral in (218) then diverges for x → 0, with the resulting equivalent functions
1
1 1
x1−a , g00 (x) ∼ − x−a , x → 0,
21−a
2
1 1
1
1
0
g(x) ∼ −
xγ+1−a = −
xγ , x → 0,
21−a
21−a
1 2
1 2p
0
γ := γ + 1 − a = + − γ = −
1 + 2κp.
2 κ
2 κ
g0 (x) ∼ −
(220)
(221)
(222)
At the other extremity of the definition interval x ∈ (0, 4], one has simply
g0 (x) ∼ 2(4 − x)1/2 4−a , g00 (x) ∼ −(4 − x)−1/2 4−a , x → 4− .
(223)
Define now
ψ0 (z, z̄) := (1 − z z̄)−β xγ g0 (x),
(224)
where γ = γ+ , β = β+ as in (211)-(212), and where we recall the special notation
x := |1 − z|2 = (1 − z)(1 − z̄) = 1 + z z̄ − z − z̄; x ∈ [0, 4], z ∈ D.
The action of the differential operator P(D) on ψ0 can be evaluated by using the
identities (103), (106), (110), and (213). After some calculation, involving in particular the identity
(z − z̄)2 = (1 − z z̄)(2 − z − z̄) − (2 + z + z̄)x,
one finds the rather simple result
P(D)[ψ0 (z, z̄)] = (1 − z z̄)−β+1 xγ−2 [A g0 (x) + B xg00 (x)] ,
(225)
κ
γ 2 − β (2 − z − z̄ − 2x),
A :=
2
κ 1 B := 2 − z − z̄ − 2x +
2(1 − z z̄) − (2 − z − z̄)x .
24−x
This can also be written as
P(D)[ψ0 (z, z̄)] = (1 − z z̄)−β+1 xγ−2 ϕ̃0 (z, z̄),
(226)
κ
ϕ̃0 (z, z̄) :=
γ 2 − β (1 − z z̄ − x)g0 (x)
(227)
2
κ 1 +xg00 (x) 1 − z z̄ − x +
(2 − x)(1 − z z̄) − x2 ,
24−x
or, by factorizing out ψ0 on the r.h.s.,
P(D)[ψ0 (z, z̄)] = ψ0 (z, z̄)(1 − z z̄)x−2 ϕ0 (z, z̄),
(228)
κ
ϕ0 (z, z̄) :=
γ 2 − β (1 − z z̄ − x)
(229)
2
g 0 (x)
κ 1 +x 0
1 − z z̄ − x +
(2 − x)(1 − z z̄) − x2 .
g0 (x)
24−x
57
Note that by formally setting g0 = 1, g00 = 0 in the above one recovers (181), as
it must. In the paragraphs below, we perform a rough analysis of the expressions
(228), (229) for z ∈ D.
• Action on the circle ∂D1/2 . On this circle (see Eq. (183) and Fig. 4), one has
1 − z z̄ − x = 0, so that for z ∈ ∂D1/2
P(D)[ψ0 (z, z̄)] = (1 − z z̄)−β+1 xγ−2 xg00 (x)
κ 1
2(1 − x)x ≤ 0,
24−x
(230)
since g00 (x) < 0 from (218).
• The x → 0, z → +1 limit. Note that
1 − z z̄ = 2<(1 − z) − |1 − z|2 = 2x1/2 cos ϕ − x,
ϕ := arg(1 − z) ∈ (−π/2, +π/2).
(231)
Hence in the limit x → 0, 1−z z̄ = O(x1/2 ), so one first gets from (228) the equivalent
κ 2
g00 (x) κ
ϕ0 (z, z̄) ∼ (1 − z z̄)
γ −β +x
1+
.
(232)
2
g0 (x)
4
One can then use the equivalents (220), and obtain
hκ
κ i
2
.
ϕ0 (z, z̄) ∼ (1 − z z̄) γ − β + (1 − a) 1 +
2
4
A calculation then yields the remarquable identity for the coefficient
κ 2
κ κ 0 2
γ + (1 − a) 1 +
= γ ,
2
4
2
(233)
(234)
where the exponent γ 0 is as defined in (222). One thus has from (229) for x → 0
2 −2 κ 0 2
P(D)[ψ0 (z, z̄)] ∼ ψ0 (z, z̄)(1 − z z̄) x
γ −β .
(235)
2
Observe then that the resulting effective coefficient, (κ/2)γ 0 2 − β, vanishes precisely
at the BS transition point p = p∗ (κ) (18), after which it is negative. Recall then
the identity (231); the equivalent (235) for z → 1 (along a ray of fixed argument
arg(1 − z) = ϕ) is positively diverging for p < p∗ (κ) and negatively diverging for
p > p∗ (κ), as follows
κ
γ02−β ,
(236)
P(D)[ψ0 (z, z̄)] ∼ ψ0 (z, z̄) 4(cos ϕ)2 x−1
2
x → 0, 1 − z = x1/2 eiϕ ,
κ 02
γ − β T 0, p S p∗ (κ).
(237)
2
58
Right at the BS transition point, the equivalent (236) vanishes. The next-order
equivalent of (232) can then be found by expanding (218) near x = 0 (the details
are ommitted here).
• The x → 4− , z → −1 limit. Note the identities
1 − z z̄ = 2<(1 + z) − |1 + z|2 = 2|1 + z| cos ϕ0 − |1 + z|2 ,
(238)
2
2
0
2
4 − x = 4 − |1 − z| = 4<(1 + z) − |1 + z| = 4|1 + z| cos ϕ − |1 + z| ,
ϕ0 := arg(1 + z) ∈ (−π/2, +π/2).
Hence in the limit x → 4− , z → −1,
4 − x ∼ 2(1 − z z̄) = O(|1 + z|),
(239)
so one first gets from (223) the equivalent to ϕ0 (229)
ϕ0 (z, z̄) ∼
16κ
, x → 4− .
(4 − x)2
(240)
This yields for (228)
P(D)[ψ0 (z, z̄)] = ψ0 (z, z̄)(1 − z z̄)x−2 ϕ0 (z, z̄)
κ 1
∼ ψ0 (z, z̄)
, z → −1, x → 4− .
24−x
(241)
• On the real axis. Consider now the expression (226), (227) on the real axis, and
write x0 := <z ∈ [−1, +1], such that now x = |1 − z|2 = (1 − x0 )2 , 1 − z z̄ = 1 − x20 .
One gets after some simplification:
P(D)[ψ0 (z, z̄)] = ψ0 (z, z̄)(1 − z z̄)x−2 × ϕ0 (x0 ),
g00 (x)
κ 1
κ 2
γ −β +x
1+
.
ϕ0 (x0 ) := 2x0 (1 − x0 )
2
g0 (x)
2 1 + x0
(242)
(243)
The function ϕ0 (x0 ) vanishes for x0 = 0, i.e., at the origin z = 0. For x0 → 1, i.e.,
z → 1, x → 0, the analysis performed in the previous paragraph shows that it also
vanishes as ϕ0 (x0 ) ∼ 2(1 − x0 ) κ2 γ 0 2 − β .
For x0 → −1+ , i.e., z → −1, x → 4− , one first has from (240) the divergent
behavior ϕ0 (x0 ) ∼ κ(1 + x0 )−2 .
A numerical analysis of the whole function between curly brackets in (243) indicates that it possesses a zero in [−1, +1] for p ≤ p∗ (κ), but not for p > p∗ (κ), which
is the case of interest here. Therefore, in that range, the overall sign of ϕ0 (x0 ) (243)
is the same as that of minus the real part of z, x0 ∈ [−1, +1], with a blowing up
ϕ0 (x0 ) → +∞ as x0 → −1. Similarly, P(D)[ψ0 (z, z̄)] < 0 for z ∈ D1/2 ∩ R, and
P(D)[ψ0 (z, z̄)] > 0 for z ∈ (D \ D1/2 ) ∩ R.
59
• The unit circle limit z z̄ → 1. If we first take the limit 1 − z z̄ → 0 in (228) and
(229), while keeping x = |1 − z|2 fixed, we get in contrast the equivalent
g00 (x)
κ x
κ 2
γ −β −x
1+
.
(244)
ϕ0 (z, z̄) ∼ x −
2
g0 (x)
24−x
For x → 4− , the whole bracket is equivalent to 4κ(4 − x)−2 , hence ϕ0 diverges
towards +∞ when z → −1 along the unit circle.
For x → 0+ , the bracket in (244) is equivalent to
−
g0
γ2 − β − x 0 .
2
g0
κ
We have seen above (see (232), (233) and (234)) that
κ 2
g00 κ
κ
lim
γ −β+x
1+
= γ 0 2 − β,
x→0 2
g0
4
2
(245)
which vanishes at the BS transition point p = p∗ (κ), and is negative for p ≥ p∗ (κ).
Hence
0
g00
κ 2
g0 κ κ 0 2
L(p, κ) := − lim
γ −β +x
= lim x
−
γ −β
x→0
x→0
2
g0
g0 4
2
κ κ 0 2
= (1 − a) −
γ −β .
(246)
4
2
We have on the unit circle, from (244) and (246), ϕ0 (z, z̄) ∼ L(p, κ) x for z →
1, z ∈ ∂D. The coefficient L(p, κ) (246) is negative right at the BS transition point
p = p∗ (κ), since its second term vanishes there to become negative for larger p. It
vanishes at the higher value
p̃(κ) := (4 + κ)/8 > p∗ (κ),
to stay positive for p > p̃(κ).
Thus, on the unit circle, and for p∗ (κ) ≤ p ≤ p̃(κ), the function ϕ0 (z, z̄), z ∈ ∂D
(244), seen as a function of x, changes sign at a finite distance along the unit circle,
going from +∞ for z = −1 to the negative equivalent ϕ0 (z, z̄) ∼ L(p, κ)|1 − z|2 for
z → +1. For p ≥ p̃(κ), the function ϕ0 (z, z̄), z ∈ ∂D, stays positive all along the
unit circle.
• Summary and conclusion of the analysis. The several cases studied above
suggest that, for p > p∗ (κ), the result P(D)[ψ0 (z, z̄)] of the action of the differential
operator on ψ0 is negative in a singly connected, crescent-like domain z ∈ C ⊂ D,
that contains the whole disk D1/2 , and the left boundary of which passes through
z = 0. A part to the right of the unit circle ∂D lies in C for p < p̃(κ), but vanishes
60
for p ≥ p̃(κ). The action is positive for z ∈ D \ C. A more thorough numerical
analysis would be needed to confirm this picture.
The main result of this section 4.2.5 is the following. Recall the two limiting
behaviors of P(D)[ψ0 (z, z̄)], respectively (235), (236), and (237) for x = |1−z|2 → 0,
P(D)[ψ0 (z, z̄)] = ψ0 (z, z̄)(1 − z z̄)x−2 ϕ0 (z, z̄),
κ
γ 0 2 − β (1 − z z̄), x → 0,
ϕ0 (z, z̄) ∼
2
1 − z z̄ = O(x1/2 ), x → 0;
(247)
(248)
and (239), (240), and (241) for z → −1, i.e., x → 4− ,
P(D)[ψ0 (z, z̄)] = ψ0 (z, z̄)(1 − z z̄)x−2 ϕ0 (z, z̄),
16κ
ϕ0 (z, z̄) ∼
, x → 4− ,
(4 − x)2
1
1 − z z̄ ∼
(4 − x), x → 4− .
2
(249)
(250)
Recall now the action (182) of the differential operator on the first solution ψ+ (z, z̄) =
(1 − z z̄)−β xγ , here with β := β+ and γ := γ+ , as in (211)-(212):
P(D)[ψ+ (z, z̄)] = ψ+ (z, z̄)(1 − z z̄)x−2 ϕ+ (z, z̄),
κ
γ 2 − β (1 − z z̄ − x).
ϕ+ (z, z̄) :=
2
(251)
Recall finally the action (190) of the operator on the logarithmically modified test
function ψ(z, z̄)`δ (z z̄) (189)
2δz z̄x−1
P(D)[ψ(z, z̄)`δ (z z̄)] = `δ (z z̄) P(D)[ψ(z, z̄)] − ψ(z, z̄)
, (252)
[− log(1 − z z̄)]
2δz z̄x−1
−2
= ψ(z, z̄)`δ (z z̄) (1 − z z̄)x ϕ(z, z̄) −
,
[− log(1 − z z̄)]
where here ψ represents either ψ0 or ψ+ , and correspondingly ϕ is either ϕ0 or ϕ+ .
Note that the two resulting functions, ϕ0 (247) and ϕ+ (251), have entirely
similar equivalent behaviors in the limit x → 0. The only difference lies in the
substitution of the parameter γ 0 (222) in ϕ0 to γ = γ+ in ϕ+ , with the result that
the coefficient (κ/2)γ 0 2 − β becomes negative for p∗ (κ) ≤ p, whereas (κ/2)γ 2 − β
was positive for p∗ (κ) ≤ p ≤ p(κ), and negative for p(κ) ≤ p. So the form of ϕ0 near
z = 1 suggests to apply the same method to ψ0 as to ψ+ in Section 4.2.3, and to use
its logarithmic modification as in (252). This would yield this time a subsolution
such that, locally, P(D)[ψ0 (z, z̄)`δ (z z̄)] < 0, instead of a super(sub)solution with a
definite sign in P(D)[ψ+ (z, z̄)`δ (z z̄)] T 0 corresponding respectively to p S p(κ).
61
However, the method used in Section 4.2.3 to prove Theorem 1.3 relied crucially on the differential action (187) on ψ+ being uniformly small for |z| → 1
outside of D1/2 . For the modified function, this allowed one to get an action
P(D)[ψ+ (z, z̄)`δ (z z̄)] with a uniform sign over a whole annulus containing ∂D. This
is turn permitted the use of the minimum (or maximum) principle, depending on
whether p ≶ p∗ (κ).
By contrast, in the case here of the second solution ψ0 , as defined by (218) and
(224), we remark that the corresponding action (249), (250) diverges towards +∞
when z → −1. This seems to pose a major obstruction to obtaining a uniformly
negative action (252) P(D)[ψ0 (z, z̄)`δ (z z̄)] < 0 for any modification ψ0 `δ of ψ0 , and
on any annulus in D whose closure contains ∂D. That in turn precludes
the
use
0
p
of the maximum principle to bound the exact solution F (z, z̄) = E |f0 (z)| from
above by ψ0 (z, z̄)`δ (z z̄). This bound, if reachable, would allow one to bound the
exact average integral means spectrum, β(p, κ) = β1 (p, κ), from above by the
√ integral
means spectrum of ψ0 `δ , i.e., that of ψ0 , namely β+ (p, κ) = 3p−(1/2)(1+ 1 + 2κp).
This, once completed by Theorem 1.3, would in turn lead to the rigorous conclusion
that these two spectra actually coı̈ncide (Statement 1.1, Eq. (15)).
4.3
Spectrum of the m-fold whole-plane SLE (m ≥ 1)
Let us first make the
Remark 4.3. The Beliaev-Smirnov average integral means spectrum β̄0 (p, κ) of the
bulk of SLEκ has a phase transition at p∗0 (κ), such that it equals successively [4]
p
4+κ
β0 (p, κ) = −p +
4 + κ − (4 + κ)2 − 8κp , 0 ≤ p ≤ p∗0 (κ), (253)
4κ
(4 + κ)2
β̂0 (p, κ) = p −
, p∗0 (κ) ≤ p,
(254)
16κ
3(4 + κ)2
p∗0 (κ) :=
.
(255)
32κ
We assert the following (non-rigorous)
(m)
Statement 4.1. For the m-fold inner whole-plane SLEκ map, defined as h0 (z) =
1/m
z f0 (z m )/z m
, with m an integer such that 1 ≤ m ≤ 3, the average integral
means spectrum is for p ≥ 0
(
)
r
2
1 1
2κp
βm (p, κ) = max β0 (p, κ), 1 +
p− −
1+
,
(256)
m
2 2
m
with a phase transition taking place when the second term on the r.h.s. of (256)
equals then supersedes the first one, at the critical point p∗m (κ) ≤ p∗0 (κ),
p
m
2
2 + 4m2 .(257)
p∗m (κ) :=
(m
+
1)(4
+
κ)
−
8m
−
4
(m
+
1)(4
+
κ)
8κ(m + 1)2
62
For m ≥ 4, the order of the two critical points p∗0 (κ) and p∗m (κ) depends on κ, and
is given by
p∗m (κ) S p∗0 (κ),
κ S κm , κm := 4
m+3
,
m−3
m ≥ 4,
(258)
such that for κ ≤ κm ,
βm (p, κ) = β0 (p, κ), p ≤ p∗m (κ),
r
2
1 1
2κp
=
1+
p− −
, p∗m (κ) ≤ p,
1+
m
2 2
m
(259)
(260)
whereas for κ ≥ κm ,
βm (p, κ) = β0 (p, κ), p ≤ p∗0 (κ),
= β̂0 (p, κ), p∗0 (κ) ≤ p ≤ p∗∗
(κ),
r m
2
1 1
2κp
p− −
, p∗∗
1+
=
1+
m (κ) ≤ p,
m
2 2
m
(261)
(262)
(263)
where p∗∗
m (κ) is the second critical point
p∗∗
m (κ) := m
κ2 − 16
,
32κ
(264)
and where the last spectrum (263) supersedes the linear spectrum (254), (262).
For p ≤ 0, the average integral means spectrum, common to all m-fold versions
of the inner or outer whole-plane SLE, is given, for −1 − 3κ/8 < p ≤ 0 by the bulk
spectrum β0 (p, κ) (253), and for p ≤ −1 − 3κ/8 by the tip-spectrum (30) [4, 33, 36]:
p
3κ
1
βm (p, κ) = βtip (p, κ) = −p +
4 + κ − (4 + κ)2 − 8κp , p ≤ −1 − . (265)
4
8
These results for βm (p, κ) satisfy in particular Makarov’s theorem 4.3 [52] for
m-fold symmetric functions:
2
2m
βm (p, κ) ≤ 1 +
p − 1 for p ≥
.
(266)
m
4+m
4.3.1
Derivation of Statement 4.1
Rather than providing here in full detail the calculation of the spectrum for the
m-fold whole plane SLE map, which is quite similar to those of Section 4.2 above,
we shall take the following shortcut, as suggested by Remark 4.1.
We now use the identities (143), (144) and (145) giving A, B and C in the m-fold
case. We first remark that the expression (145) for C does not depend on m, hence
63
the standard spectrum, as given by β = κα2 /2 and C = − κ2 α(α − 1) + 2α − p = 0,
coı̈ncides with the BS spectrum β0 (p) (for the choice of the negative branch solution
β0− (p) to C = 0). This is expected, since this part of the spectrum should correspond
to the fine multifractal structure of the bulk of the SLE curve, which should stay
invariant under any m-fold transform.
The spectrum corresponding to the unbounded nature of the m-fold whole-plane
SLE can now be obtained by setting A = 0 in (143), and using again β = κα2 /2 − C
together with C (145). This gives the two solutions
p
(m)
(267)
α = γ± (p, κ) := κ−1 1 ± 1 + 2κp/m ,
κ ±
2
(m)
+ 1 p − γm
β = β± (p, κ) :=
m
2
r
2
1
p
=
+1 p−
1 ± 1 + 2κ
.
(268)
m
2
m
Thanks to the universal spectrum for m-fold symmetric analytic functions, as given
(m)
by Makarov’s theorem 4.3, the negative branch β− is clearly excluded, while the
(m)
positive one β+ satisfies the universal bound (266). The transition point where
(m)
β+ (p, κ) = β0 (p, κ) is given by p = p∗m (κ) in Eq. (257).
For p ≤ p∗m (κ), and if the transition point p∗m (κ) ≤ p∗0 (κ) (255), where the BS
(m)
spectrum changes to the linear spectrum (254), one has β0 (p, κ) ≥ β+ (p, κ), hence
βm (p, κ) = β0 (p, κ), while for p ≥ p∗m (κ), the unbounded integral means spectrum
(m)
(m)
β+ (p, κ) dominates, hence βm (p, κ) = β+ (p, κ).
The alternative inequality p∗0 (κ) ≤ p∗m (κ) arises only for m ≥ 4 and for κ ≥ κm
(258). In this case, the phase transition at p∗0 (κ) to the linear piece (254) of the BS
spectrum appears before the phase transition from β0 (p, κ) to the m-fold unbounded
(m)
spectrum β+ (p, κ) happens at p∗m (κ). The latter transition therefore takes place at
(m)
the second phase transition point p∗∗
m (κ) (264), where β̂0 (p, κ) intersects β+ (p, κ).
We thus expect the sequence of spectra (261), (262), and (263) to take place for
m ≥ 4, κ ≥ κm . This concludes the (non-rigorous) derivation of statement 4.1.
4.3.2
Proof of Theorem 1.4
Proof. A rigorous proof of Theorem 1.4 for general m can be achieved in the same
manner as in Section 4.2 above for the m = 1 case. Using the differential operator
(141)
κ
ζ +1
ζ +1
Pm (D) := − (ζ∂ζ − ζ∂ζ )2 +
ζ∂ζ +
ζ∂ζ
2
ζ −1
ζ −1
p m−1 m−1
m
m
+
+
−
−
+2 ,
m 1−ζ
(1 − ζ)2 (1 − ζ)2
1−ζ
64
(269)
instead of (101) (or the cylindrical coordinate version thereof instead of (152)) in
the analysis of Section 4.2, we search for approximate solutions of the form
ψ(ζ, ζ̄) := (1 − ζ ζ̄)−β xγ ,
(270)
with here
ζ := z m , x := (1 − ζ)(1 − ζ̄) = (1 − z m )(1 − z̄ m ).
Since ψ(ζ, ζ̄) = (1 − ζ ζ̄)−β ϕγ (ζ)ϕγ (ζ̄), with ϕγ (ζ) = (1 − ζ)γ , the same algebra as in
(142) in Section 4.2.2 yields, for arbitrary values of β, γ, the analogue of the action
(170)
Pm (D)ψ(ζ, ζ̄) =
ψ(ζ, ζ̄)x−1
(271)
1 − ζ ζ̄
2
+1 −2 ,
× (κγ − 2β)ζ ζ̄ − Am (1 − ζ ζ̄ − x) + C (1 − ζ ζ̄)
x
where Am := A is given by (143) and C by (145), in conjunction with (142):
Pm (∂)[ϕγ ] = Am ϕγ + Bm ϕγ−1 + Cϕγ−2 ,
p
κ
Am = − γ 2 + γ + ,
2
m
Bm = −Am − C,
κ
C = − γ(γ − 1) + 2γ − p.
2
(m)
(272)
(273)
(274)
(275)
(m)
Here we choose β := β+ and γ := γ+ , and write ψ = ψ+ , as in Eqs. (267) and
(268), such that Am = 0 and C = κγ 2 /2−β. The action (271) then simply becomes,
as in (182),
κ
2
Pm (D)[ψ(ζ, ζ̄)] = = ψ(ζ, ζ̄)
γ − β (1 − ζ ζ̄)(1 − ζ ζ̄ − x)x−2 . (276)
2
In complete analogy to Eqs. (187) and (188), we have, with the special point pm (κ)
defined as in (146),
p < pm (κ) : Pm (D)[ψ+ (ζ, ζ̄)] > 0, ζ ∈ D1/2 , Pm (D)[ψ+ ] < 0, ζ ∈ D \ D1/2 ;
p = pm (κ) : Pm (D)[ψ+ (ζ, ζ̄)] = 0, ζ ∈ D;
p > pm (κ) : Pm (D)[ψ+ (ζ, ζ̄)] < 0, ζ ∈ D1/2 Pm (D)[ψ+ ] > 0, ζ ∈ D \ D1/2 .
A modification by the logarithmic factor (189) of ψ+ into ψ+ (ζ, ζ̄)`δ (ζ ζ̄), yields the
same conclusions as in Section 4.2.3: there exist open annuli Am (ri ) := {ζ : ri <
|ζ| < 1} = D \ D(ri ), i = 1, 2, whose boundary includes ∂D, and where one has
respectively (for a specific sign of δ chosen appropriately to each case):
• for p < pm (κ), ψ+ `δ is a supersolution with Pm (D)[ψ+ (ζ, ζ̄)`δ (ζ ζ̄)] > 0 for ζ ∈
Am (r1 );
65
• for p > pm (κ), ψ+ `δ is a subsolution with Pm (D)[ψ+ (ζ, ζ̄)`δ (ζ ζ̄)] < 0 for ζ ∈
Am (r2 );
• for p = pm (κ), Pm (D)[ψ+ (ζ, ζ̄)] = 0, ζ ∈ D, so that ψ+ (ζ, ζ̄) = F (ζ, ζ̄) = (1 −
ζ ζ̄)−β |1 − ζ|2γ is the exact solution of Theorem 3.8 with parameters (146): γ =
(m)
(m)
γ+ (pm (κ), κ) = αm (κ) and β = β+ (pm (κ), κ) = καm (κ)2 /2.
We then follow the same method as above [3, 4]. The operator Pm (D), when
written in polar coordinates, is parabolic. Using in each of the two annuli where
Pm (D)[ψ+ `δ ] has a definite sign, respectively, the minimum principle, and the maximum principle ([27], Th. 7.1.9), yields
Proposition 4.3. There exist two positive constants ci , i = 1, 2, such that
where ζ = z m
c1 ψ+ `δ < F, ζ ∈ Am (r1 ), p < pm (κ),
(277)
ψ+ = F, ζ ∈ D, p = pm (κ),
F < c2 ψ+ `δ , ζ ∈ Am (r2 ), p > pm (κ).
(278)
(m)
(m)
and F = F (ζ, ζ̄) := E |(h0 )0 (z)|p , with h0 (z) := [f0 (z m )]1/m .
From the inequality (277) (resp. (278)), we therefore conclude that the spectrum
associated with ψ+ or ψ+ `δ ,
r
1 1
2κp
2
(m)
p− −
1+
,
(279)
β+ (p, κ) = 1 +
m
2 2
m
(m)
is, for p ≤ pm (κ), (resp. for p ≥ pm (κ)), a lower bound β+ ≤ βm (resp. upper
(m)
bound β+ ≥ βm ) to the exact average integral means spectrum βm (p, κ) of the
m-fold inner whole-plane SLEκ .
Recall then that the BS average integral means spectrum β̄0 (p, κ) (253)-(254)
(m)
of Remark 4.3 becomes smaller than the spectrum β+ (p, κ) (279) at the transition point p = p∗m (κ) (257) [for 1 ≤ m ≤ 3, ∀κ, or for m ≥ 4, κ ≤ κm (258)],
or at the transition point p = p∗∗
m (κ) (264) [for m ≥ 4, κ ≥ κm ]. Observe that
these transition values are both smaller than the special point pm (κ). We thus conclude that βm (p, κ) = β̄0 (p, κ) before these transition points, whereas necessarily
(m)
βm (p, κ) ≥ β+ (p, κ) > β̄0 (p, κ) after them. At the higher special value p = pm (κ),
(m)
we know that the two spectra βm (p, κ) and β+ (p, κ) coı̈ncide. For p > pm (κ), the
(m)
inequality is reversed: βm (p, κ) ≤ β+ (p, κ). This concludes the proof of Theorem
1.4.
4.4
4.4.1
Integral means spectrum and derivative exponents
Motivation
In this section, we (heuristically) explain the striking relationship between the packing spectrum (36) and the whole-plane average integral means spectrum (37) after
66
the phase transition that takes place at p = p∗ (κ) (18). (See also Remark 1.4.)
The average integral means spectrum (8) involves evaluating, for the whole-plane
SLE map f0 (z), the integral
Z
Ip (r) :=
E [|f00 (rz)|p ] |dz|,
(280)
∂D
on a circle of radius r < 1 concentric to ∂D, and taking the limit for r → 1−
β(p) = lim sup
r→1
log Ip (r)
.
− log(1 − r)
(281)
If Ip (r) has a power law behavior, an alternative definition of β(p) would be such
that
r→1
(282)
(1 − r)β(p) Ip (r) 1.
To understand why the average integral means spectrum, for p ≥ p∗ (κ) (18), crosses
over to the special whole-plane form (37), one should consider that the integrand in
(280) behaves more like a distribution for p large enough. Then, the circle integral
(280) concentrates in the vicinity of the pre-image point z0 := f0−1 (∞) ∈ ∂D, which
is sent to infinity by the unbounded whole-plane SLE map f0 (Fig. 1). To see this
and the relation to the packing spectrum, we first need to recall the relation of our
inner whole-plane SLE to standard radial SLE.
4.4.2
Radial and whole-plane SLE
Let us consider the standard inner radial SLEκ process gt in the unit disk D [63],
satisfying the stochastic differential equation
∂t gt (w) = gt (w)
√
λ(t) + gt (w)
, λ(t) = ei κBt , t ≥ 0.
λ(t) − gt (w)
It is defined for w ∈ D \ Kt , where (Kt , t ≥ 0) is a random increasing family of
subsets (hulls) of the unit disk that grows towards the origin 0 (Fig. 6). The map gt
is the unique conformal map from D \ Kt onto D, such that gt (0) = 0 and gt0 (0) = et .
Denote by gt−1 (z), t ≥ 0, z ∈ D, the inverse map of gt , which maps D to D \ Kt
(Fig. 6). It is such that gt−1 (0) = 0 and (gt−1 )0 (0) = 1/gt0 (0) = e−t . It also has the
same law as the continuation to negative times, g−t , of the forward radial map gt .
Consider now the inner whole-plane map ft (z) as defined in (2) for t ≥ 0, z ∈ D, its
inverse map ft−1 , and the whole-plane map at t = 0, f0 (z). Define
ϕt (z) := ft−1 ◦ f0 (z), z ∈ D.
(283)
We then have the identities in law [44, 60]
(law)
(law)
ϕt (z) = g−t (z) = gt−1 (z).
67
(284)
g t (w )
w
rt
Lt
g t ( At )
At
0
ID
e ït
0
ID \ Kt
gï1
t (z )
Figure 6: Inverse Schramm-Loewner map z 7→ gt−1 (z) from D to the slit domain
D \ Kt , where Kt is the SLEκ hull (here a single curve for κ ≤ 4). The distance from
the SLE tip to the origin is of order e−t when t → +∞. The length Lt := |gt (At )| of
the image of the boundary set At := ∂D \ Kt gives the harmonic measure Lt /2π of
At as seen from 0 in D \ Kt . The inner circle radius rt is chosen so that 1 − rt = Lt .
As already mentioned in Section 3.2.1 (see also [44]), the limit for t → +∞ of
(law)
(law)
et ϕt (z) = et g−t (z) = et gt−1 (z) exists, and has the same law as the inner wholeplane process f0 (z):
f˜t (z)
f0 (z)
:=
(law)
=
et gt−1 (z), z ∈ D,
lim f˜t (z) = lim et gt−1 (z).
t→+∞
t→+∞
(285)
(286)
In the limit t → +∞ of the radial inverse SLE map et gt−1 , the boundary circle et ∂D
is pushed back to infinity, while the limit of hulls (et Kt )t→+∞ becomes the wholeplane SLE hull (e.g., for κ ≤ 4 the single slit γ([0, ∞)) in Fig. 1). Since the tip
of gt−1 (∂D) is at distance of order e−t from 0, the limit of the tip of et gt−1 (∂D) for
t → +∞ stays at a finite distance from 0, as does the tip f0 (1) = γ(0) (Fig. 1 and
Fig. 6).
4.4.3
Packing and derivative exponents
Let us briefly recall Lemma 3.2 in Ref. [42]:
68
Lemma 4.1. Let
At := ∂D \ Kt ,
which is either an arc on ∂D or At = ∅. Let s ≥ 0, and set
p
s
1 ν = ν(s, κ) := +
κ − 4 + (4 − κ)2 + 16κs .
2 16
(287)
Assume κ > 0 and s > 0. Let H(θ, t) denote the event {w = exp(iθ) ∈ At }, and set
s
(288)
F(θ, t) := E gt0 exp(iθ) 1H(θ,t) ,
p
κ − 4 + (4 − κ)2 + 16κs
q = q(s, κ) := Uκ−1 (s) =
,
(289)
2κ
q
h∗ (θ, t) := exp(−ν t) sin(θ/2) .
Then there is a constant c > 1 such that
∀t ≥ 1, ∀θ ∈ (0, 2π),
h∗ (θ, t) ≤ F(θ, t) ≤ c h∗ (θ, t),
(290)
which we denote by h∗ (θ, t) F(θ, t).
By conformal invariance, the harmonic measure from 0 of the boundary arc At
in the slit domain Dt := D \ Kt is Lt /2π, where Lt is the length of the arc gt (At ).
Let us then also recall Theorem 3.3 in [42]:
Theorem 4.4. Suppose the κ > 0 and s ≥ 1. Then, when t → +∞,
E [(Lt )s ] exp(−ν t).
Notice that Lemma 4.1 and Theorem 4.4 taken together strongly suggest that
Lt for w ∈ At and t → +∞.
Let us now use (286), and replace the whole-plane map f0 (z) by its large time
equivalent in law, f˜t (z) = et gt−1 (z) (285), taken for some large time t. The domain
which is sent far away from the origin by this map f˜t is the subset gt (At ) ⊂ ∂D of
the unit circle, as well as its immediate vicinity in D (see Fig. 6). This corresponds
in the image domain D \ Kt to w ∈ At . Define then the restricted boundary integral:
Z
Ip (t) :=
ept |(gt0 (w)|s |dw|
(291)
At
Z 2π
=
ept |(gt0 eiθ |s 1H(θ,t) dθ
|gt0 (w)|
0
s = s(p, κ) = β(p, κ) + 1 − p,
(292)
where β(p, κ) is given by (37) and s(p, κ) by (38). This choice for s is precisely the
one that insures that ν (287) equals
ν(s(p, κ), κ) = p.
69
(293)
From (290) in Lemma 4.1 we have for w ∈ At the asymptotic behavior for large t
s
q
E gt0 (w) 1H(θ,t) exp(−ν t) sin(θ/2) ,
so that
E Ip (t) Z
2π
sinq (θ/2)dθ.
(294)
0
This integral converges as Eqs. (38), (287), and (293) imply that
s
s
1 p
κ
q(s, κ) = ν(s, κ) − = p − =
1 + 2κp − 1 ≥ 0.
8
2
2
4
The integral (291) over At can be mapped back via the gt map to the subset gt (At )
of the unit circle and equals
Z
0
Ip (t) =
ept |(gt−1 (z)|p−β(p) |dz|, z = gt (w),
(295)
gt (At )
Z
0
=
|f˜t0 (z)|p |(gt−1 (z)|−β(p) |dz|.
gt (At )
Consider then the set of sub-arc integrals
Z
0
Ip (C, t) :=
|f˜t0 (z)|p |(gt−1 (z)|−β(p) |dz|,
(296)
C
Ip (C, t) ≤ Ip (t), ∀C
gt (At ).
(297)
Using Schwarz’s reflection principle, the function gt−1 (z) (therefore its blow-up f˜t (z) =
et gt−1 (z)) can be analytically extended, outside of the unit disk D, by inversion with
respect to the unit circle of any angular sector spanned by a strict sub-arc C gt (At ).
Koebe’s theorem then implies in all these angular sub-sectors the uniformly bounded
behavior
f˜0 (rz) g −1 (rz) t
≤ C, r ≤ 1, ∀z ∈ C gt (At ) ⊂ ∂D,
(298)
C −1 ≤ t 0
=
f˜t (z) gt−1 (z) where the constant C depends on the sub-arc C of gt (At ), and may go to infinity
when C → gt (At ).
By Koebe’s bounds (298), one can extends the boundary integral (296) to the
interior of D:
Z
0
Ip (C, t) Ip (r C, t) := |f˜t0 (rz)|p |(gt−1 (rz)|−β(p) |dz|, ∀r ≤ 1, ∀C gt (At ). (299)
C
70
Introduce now the time-dependent (random) radius rt
rt := 1 − Lt ; rt → 1− , Lt → 0, t → +∞.
(300)
In the boundary arc w ∈ gt−1 (C) At , and z = gt (w) ∈ C gt (At ), we have seen
0
that the derivative tends to zero uniformly as |gt0 (w)| = |(gt−1 (z)|−1 Lt = 1 − rt ,
for t → +∞. For the particular choice of radius r = rt , the equivalence (299) can
then be rewritten as
Z
β(p)
(301)
Ip (C, t) Ip (rt C, t) (1 − rt )
|f˜t0 (rt z)|p |dz|, ∀C gt (At ).
C
As suggested in Section 4.4.1 above in the case of the whole-plane map f0 , and
for p ≥ p∗ (κ), one assumes that the similar integral, extended to the whole circle
of radius rt < 1, is dominated by the localized integral (301) for t → +∞. This
condensation of the integral’s support is precisely the signal of the onset of the
transition from the standard SLE bulk spectrum β0 (p, κ) (254) to the whole-plane
spectrum β(p, κ) (37). We therefore expect for p ≥ p∗ (κ)
Z
β(p)
Ip (rt C, t) (1 − rt )
|f˜t0 (rt z)|p |dz|.
(302)
∂D
From Eqs. (297), (301) and (302), and from the finite expectation result (294), one
therefore concludes that, in expectation,
Z
β(p)
0
p
˜
E (1 − rt )
|ft (rt z)| |dz| 1, t → +∞.
(303)
∂D
Since the random radius rt → 1, this equivalence is (formally) very similar to the
equivalence (282) above, which can serve as a heuristic definition of the average
integral means spectrum. This strongly suggests why the average integral means
spectrum β(p, κ) (37), which is specific to the unbounded whole-plane SLE map,
is intimately related, via the packing spectrum s(p, κ) = β(p, κ) − p + 1 (36), to
the derivative exponents (287), as derived by Lawler, Schramm and Werner in Ref.
[42]: the derivative exponent p = ν(s, κ) is the inverse function of the unbounded
whole-plane packing spectrum s(p, κ).
5
5.1
5.1.1
Appendices
Appendix A: A brief history of Bieberbach’s conjecture
Proof for n = 2 (Bieberbach [7], 1916)
First, let us introduce the normalized, so-called schlicht class of univalent functions
S = {f : D → C holomorphic and injective ; f (0) = 0, f 0 (0) = 1}.
71
The Bieberbach conjecture is clearly equivalent to |an | ≤ n, n ≥ 2 for f ∈ S. A
related class of normalized functions is
)
(
∞
X
−n
bn z at ∞ .
Σ = f : ∆ = C\D → C holomorphic and injective ; f (z) = z +
n=0
The mapping f 7→ F , where F (z) = 1/f (1/z), is clearly a bijection from S onto
Σ0 , the subclass of Σ consisting of functions that do not vanish in ∆. A simple
application of the Stokes formula shows that if f ∈ Σ then, denoting by |B| the
Lebesgue measure (area) of the Borelian subset B of the plane,
X
|C\f (∆)| = π 1 −
n|bn |2 .
n≥1
Since the area is a positive quantity, a consequence of this equality is that |b1 | ≤ 1.
But applying this inequality directly to the function F , image in Σ0 of f ∈ S, does
bring anything conclusive. Bieberbach’s idea was then to apply this inequality to
an odd function in S.
Let f ∈ S then z 7→ f (z)/z does not vanish in the disc and thus it has a
unique holomorphic square root g which is equal to 1 at 0. Then, h(z) = zg(z 2 ),
such that f (z 2 ) = h(z)2 , is still in class S but is moreover odd. This establishes
a bijection (f 7→ h) between S and the set of odd functions in S. Now, if f (z) =
z + a2 z 2 + a3 z 3 + . . . belongs to class S, then h(z) = z + 21 a2 z 3 + O(z 5 ) and the
associated H ∈ Σ satisfies H(z) = 1/h(1/z) = z − a2z2 + · · · By the area proposition,
|a2 | ≤ 2.
Remark 5.1. The fact that |a2 | is bounded above for functions in class S implies
(see [56]) that the class S is compact.
Remark 5.2. As a corollary, one can state a weak form of the Bieberbach conjecture, namely that for each nP≥ 2 there exists a positive constant Cn < +∞ such that
for any f ∈ S, f (z) = z + n≥2 an z n , then |an | ≤ Cn .
5.1.2
Proof for n = 3 (Loewner [50], 1923)
Replacing f (z) by f (rz) with r < 1 but close to 1, one sees that it suffices to
prove the estimate for conformal mappings onto smooth Jordan domains containing
0. Consider such a domain Ω and let γ : [0, t0 ] → C be a parametrization of its
boundary. Introduce then Γ : [0, ∞) → C, a Jordan arc joining γ(0) = γ(t0 ) to ∞
inside the outer Jordan component. We then define
Λ(t) := γ(t), 0 ≤ t ≤ t0 ; Λ(t) := Γ(t − t0 ), t ≥ t0 ,
and define for t > 0,
Ωt = C\Λ([t, ∞)).
72
The domain Ωt is a simply connected domain containing 0 and we can thus consider
its Riemann mapping ft : D → Ωt , ft (0) = 0, ft0 (0) > 0. By the Caratheodory
convergence theorem, ft converges as t → 0 to f , the Riemann mapping of Ω. We
may assume without loss of generality that f 0 (0) = 1 and, by a change of time t if
necessary, that ft0 (0) = et .
The key idea of Loewner is to observe
that the sequence of domains Ωt is increas∂ft
∂ft
ing, which translates into < ∂t /z ∂z > 0 or, equivalently, that the same quantity
is the Poisson integral of a positive measure, actually a probability measure because
of the choice of parametrization ft0 (0) = et . Now, the fact that the domains Ωt are
slit domains implies that for every t this probability measure must be, on the unit
circle, the Dirac mass at λ(t) = ft−1 (Λ(t)). Even if this is not needed in Loewner’s
proof, it is worthwhile to notice that λ is a continuous function. The process Ωt is
then driven by the function λ, in the sense that (ft ) satisfies the Loewner differential
equation
∂ft
∂ft λ(t) + z
=z
.
∂t
∂z λ(t) − z
(304)
To finish Loewner’s proof, one extends both sides of the last equation as power
series, with ft (z) = et (z + a2 z 2 + a3 z 3 + · · · ), and simply identifies the coefficients,
as was done in Section 2. This leads to:
a˙2 − a2 = 2λ,
2
a˙3 − 2a3 = 4a2 λ + 2λ .
As seen above (Eqs. (55) & (56)), this is solved by
t
∞
Z
λ(s)e−s ds,
a2 (t) = −2e
t
a3 (t) = 4e
2t
Z
∞
−s
λ(s)e ds
2
2t
Z
− 2e
∞
2
e−2s λ (s)ds.
t
t
The first equation gives a new proof that |a2 | ≤ 2|a1 | = 2. For a3 , by considering
e−iα f (eiα z), one remarks that it suffices to prove that <(a3 ) ≤ 3. To this aim, write
λ(s) = eiθ(s) . The Cauchy-Schwarz inequality,
Z
et
∞
−s
e
2
Z
t
cos θ(s)ds ≤ e
t
t
73
∞
e−s cos2 θ(s)ds ,
gives
<(a3 ) = 4e
2t
Z
2
∞
−s
e
cos θ(s)ds
t
2t
Z
2
∞
−s
2t
Z
e sin θ(s)ds − 2e
− 4e
t
Z ∞
≤4
et−s − e2(t−s) cos2 θ(s)ds + 1
Zt ∞
≤4
et−s − e2(t−s) ds + 1 = 3.
∞
e−2s cos 2θ(s)ds
t
t
5.1.3
The Bieberbach conjecture after Loewner
The next milestone after the 1923 theorem by Loewner is the proof in 1925 by
Littlewood [47] that in class S, |an | ≤ en. In 1931, Dieudonné [13] proved the
conjecture for functions with real coefficients. In 1932, Littlewood and Paley [48]
proved that the coefficients of an odd function in S are bounded by 14, and they
conjectured that the best bound is 1, a conjecture that implies Bieberbach’s. This
conjecture was disproved in 1933 by Fekete and Szegö [28] for n = 5. In 1935,
Robertson [59] stated the weaker conjecture
n
X
|a2k+1 |2 ≤ n,
k=1
which also implies the Bieberbach conjecture. The next milestone was due in the
sixties to Lebedev and Milin [46]. It had already been observed by Grunsky [31] in
1939 that the logarithmic coefficients γn defined by
log[f (z)/z] = 2
∞
X
γn z n
n=1
can easily be estimated. Lebedev and Milin [46] showed, through three inequalities,
how to pass from those estimates to estimates for f . This allowed Milin [54] to prove
that |an | ≤ 1.243 n. He then stated what has become known as Milin conjecture:
n X
m
X
k|γk |2 − 1/k ≤ 0.
m=1 k=1
It should be noticed that γn = 1/n for the Koebe function but the stronger conjecture |γn | ≤ 1/n is false, even as an order of magnitude. It happens that Milin ⇒
Robertson ⇒ Bieberbach, and De Branges actually proved the Milin conjecture.
74
5.2
5.2.1
Appendix B
Quadratic third order coefficient
For calculations involving a3 as given by (56), and in order to avoid repetitions, we
will compute at once E(|a3 − µa22 |2 ), where µ is a real constant. We write
e−4t |a3 − µa22 |2 = 16(1 − µ)2 I1 − 16(1 − µ)<I2 + 4I3 ,
where
Z
∞
Z
∞
Z
∞
Z
I1 =
t
Z
∞
t
Z
∞
t
Z
∞
I3 =
t
e−(s1 +s2 +s3 +s4 ) λ(s1 )λ(s2 )λ(s3 )λ(s4 )ds1 ds2 ds3 ds4 ,
t
e−(s1 +s2 +2s3 ) λ(s1 )λ(s2 )λ(s3 )2 ds1 ds2 ds3 ,
I2 =
Zt ∞ Zt ∞
∞
t
e−2(s1 +s2 ) λ(s1 )2 λ(s2 )2 ds1 ds2 .
t
From now on, we set the parameter t = 0 in the above formulae. The computation
of I3 follows the same lines as that in Proposition 2.1 and we find
1
.
E(I3 ) = <
2(2 + η2 )
To compute E(I2 ) we use the strong Markov property. First, we may write by
symmetry
Z ∞ Z ∞ Z ∞
I2 = 2
e−(s1 +s2 +2s3 ) ei(Ls3 −Ls1 ) ei(Ls3 −Ls2 ) ds1 ds2 ds3 ;
s1 =0
s2 =s1
s3 =0
we cut this integral into I2 = 2(I2,1 + I2,2 + I2,3 ), where in I2,1 (resp. in I2,2 , I2,3 ), s3
lies in [0, s1 ] (resp. in [s1 , s2 ], [s2 , ∞)). For I2,1 , write
ei(Ls3 −Ls1 ) ei(Ls3 −Ls2 ) = e−2i(Ls1 −Ls3 ) e−i(Ls2 −Ls1 ) ,
so that the Markov property can be used to get its expectation as e−η2 (s1 −s3 ) e−η1 (s2 −s1 ) .
From this, the value of E(I2,1 ) easily follows as
E(I2,1 ) =
1
.
4(1 + η1 )(2 + η2 )
Similar considerations lead to
E(I2,2 ) =
1
1
, E(I2,3 ) =
.
4(1 + η1 )(3 + η1 )
4(2 + η2 )(3 + η1 )
75
By combining these computations we get
1
1
1
+
+
.
<E(I2 ) = <
2(1 + η1 )(2 + η2 ) 2(1 + η1 )(3 + η1 ) 2(2 + η2 )(3 + η1 )
The computation of I1 follows the same lines. First, by symmetry,
Z ∞Z ∞Z ∞Z ∞
e−(s1 +s2 +s3 +s4 ) ei(Ls3 −Ls1 ) ei(Ls4 −Ls2 ) ds1 ds2 ds3 ds4 .
I1 = 4
0
s1
0
s3
We then split this integral into the sum of six pieces, respectively associated with
the domains (I) s3 < s4 < s1 < s2 ; (II) s3 < s1 < s4 < s2 ; (III) s3 < s1 < s2 < s4 ;
(IV) s1 < s3 < s4 < s2 ; (V) s1 < s3 < s2 < s4 ; (VI) s1 < s2 < s3 < s4 .
Clearly, the respective contributions of (I) and (VI), (II) and (V), (III) and
(IV), are complex conjugate of each other. The same arguments as above give, in a
short-hand notation,
1
,
4(1 + η1 )(2 + η2 )(3 + η1 )
1
1
E(II) =
, E(III) =
.
8(1 + η1 )(3 + η1 )
8(1 + η1 )(3 + η1 )
E(I) =
Altogether, we get
2
1
1
E(I1 ) = <
+
+
.
(1 + η1 )(2 + η2 )(3 + η1 ) (1 + η1 )(3 + η1 ) (1 + η1 )(3 + η1 )
We may now state the following result, which is also quoted in the main text:
Proposition 5.1. If µ is a real coefficient then
E(|a3 − µa22 |2 ) =
16(1 − µ)(2 + η1 )
2
8(1 − µ)(1 − 2µ)
16(1 − µ)2 (4 + η2 )
<
−
+
+
.
(1 + η1 )(2 + η2 )(3 + η1 ) (1 + η1 )(2 + η2 )(3 + η1 ) 2 + η2
(η1 + 1)(η1 + 3)
In the real η case:
E(|a3 − µa22 |2 ) =
32(1 − µ)2 (3 + η2 ) − 8(1 − µ)(6 + 2η1 + η2 ) + 2(1 + η1 )(3 + η1 )
.
(1 + η1 )(2 + η2 )(3 + η1 )
In the SLE case, ηj = κ2 j 2 :
E(|a3 −
µa22 |2 )
(108 − 288µ + 192µ2 ) + (88 − 208µ + 128µ2 )κ + κ2
=
.
(1 + κ)(2 + κ)(6 + κ)
76
5.2.2
Higher orders
Using dynamic programing, we performed computations (formal up to n = 8 and
numerical up to n = 19) on a usual computer. Here are the results for a3 , a4 and
a5 in the LLE case:
E(|a3 |2 ) =
E |a4 |2
=
+
E |a5 |2
=
+
+
Q =
+
−
+
2(η1 − 1)(η1 − 3)
3!22
+
;
(η1 + 1)(η1 + 3) (η1 + 1)(η1 + 3)(η2 + 2)
4!23
4(η1 − 1)(η1 − 3)η2 (η2 − 4)(η1 + 3)
+
(η1 + 1)(η1 + 3)(η1 + 5) 3(η1 + 1)(η1 + 3)(η1 + 5)(η2 + 2)(η2 + 4)(η3 + 3)
32(η1 − 1)(η1 − 3)
;
(η1 + 1)(η1 + 3)(η1 + 5)(η2 + 2)(η2 + 4)
5!24
(η1 + 1)(η1 + 3)(η1 + 5)(η1 + 7)
4(η1 − 1)(η1 − 3)η2 (η2 − 4)(η1 + 3)(η3 + 1)(η3 − 5)(η1 + 3)(η1 + 5)(η2 + 4)
3(η1 + 1)(η1 + 3)(η1 + 5)(η1 + 7)(η2 + 2)(η2 + 4)(η2 + 6)(η3 + 3)(η3 + 5)(η4 + 4)
(η1 − 1)(η1 − 3)Q
(η1 + 1)(η1 + 3)(η1 + 5)(η1 + 7)(η2 + 2)(η2 + 4)(η2 + 6)(η3 + 3)(η3 + 5)
4
(24η12 η22 + 9η12 η2 η32 + 72η12 η2 η3 + 39η12 η2 + 36η12 η32 + 288η12 η3 + 520η12 + 19η1 η23 η3
3
77η1 η23 + 56η1 η22 η3 + 472η1 η22 − 36η1 η2 η32 − 816η1 η2 η3 − 3660η1 η2 − 144η1 η32
1152η1 η3 − 2160η1 + 75η23 η3 + 285η23 + 348η22 η32 + 2952η22 η3 + 6420η22 + 3507η2 η32
26184η2 η3 + 43245η2 + 8460η32 + 67680η3 + 126900).
In each expression for E(|an |2 ), and after the first term there, notice the presence of
the common factors (η1 − 1)(η1 − 3) in denominators. The first term itself equals 1
for η1 = 3 (or κ = 6), or equals n for η1 = 1 (or κ = 2).
We end this Appendix with the results for a4 to a8 in the SLE case, and their
77
graphs in terms of κ:
2
E |a4 |
8 κ5 + 104κ4 + 4576κ3 + 18288κ2 + 22896κ + 8640
=
;
9
(κ + 10)(3κ + 2)(κ + 6)(κ + 1)(κ + 2)2
E |a5 |2 = (27κ8 + 3242κ7 + 194336κ6 + 6142312κ5 + 42644896κ4
+ 119492832κ3 + 153156096κ2 + 87882624κ + 18144000)
/[36(κ + 14)(3κ + 2)(κ + 10)(2κ + 1)(κ + 6)(κ + 3)(κ + 1)(κ + 2)2 ] ;
2
E |a6 |2 =
(216κ10 + 29563κ9 + 2062556κ8 + 90749820κ7 + 2277912280κ6
225
+ 16419864848κ5 + 50825787744κ4 + 76716664128κ3
+ 58263304320κ2 + 21233664000κ + 2939328000)
/[(κ + 18)(3κ + 2)(κ + 14)(2κ + 1)(κ + 10)(κ + 6)(5κ + 2)
(κ + 3)(κ + 1)(κ + 2)2 ] ;
E |a7 |2 =
1
(27000κ15 + 4479353κ14 + 373838334κ13 + 20594712527κ12
8100
+ 787796136854κ11 + 19121503739240κ10 + 221861771218136κ9
+ 1386550697705712κ8 + 5130607642056896κ7 + 11854768997862912κ6
+ 17547915006086400κ5 + 16725481436226816κ4 + 10110569026936320κ3
+ 3711483045734400κ2 + 749049576192000κ + 63371911680000)
/[(κ + 22)(3κ + 1)(5κ + 2)(κ + 18)(2κ + 1)(κ + 14)(3κ + 2)
(κ + 10)(κ + 6)(κ + 5)(κ + 3)(κ + 1)2 (κ + 2)3 ] ;
E |a8 |2 =
2
(729000κ18 + 143757261κ17 + 14031668642κ16 + 906444920407κ15
99225
+ 42715714646750κ14 + 1476227672190480κ13 + 34674813906653712κ12
+ 471116720002819536κ11 + 3802657434377773600κ10
+ 19218418658636100992κ9 + 63191729416067875840κ8
+ 138392538501661946112κ7 + 204258207932541043200κ6
+ 203508494170475323392κ5 + 135640094878259859456κ4
+ 59063686024095313920κ3 + 16005106174366310400κ2
+ 2435069931098112000κ + 158176291553280000)
/[(7κ + 2)(5κ + 2)(κ + 26)(3κ + 1)(κ + 22)(2κ + 1)(κ + 18)(κ + 14)
(3κ + 2)(κ + 10)(κ + 5)(κ + 3)(κ + 6)2 (κ + 1)2 (κ + 2)3 ] .
78
These results call for two observations:
–Somehow surprisingly, all the coefficients of the polynomial expansions in κ are
positive.
–For κ → ∞ (or η → ∞), these expectations vanish as κ−1 .
5.3
Appendix C: A proof of Theorem 4.2
Proof. The proof closely follows the steps of that of Feng-McGregor’s theorem. We
start with the computation of h0 :
h0 (z) = zf 0 (z 2 )(f (z 2 ))−1/2 .
We may then write, putting ρ = r2 ,
Z
Z 2π
0
iθ p
|h (re )| dθ ≤
2π
0
0
|f 0 (ρeiθ )|p
dθ.
|f (ρeiθ )|p/2
Consider now two positive reals a, b such that a − b = 1 and fix 0 < p < 2. By
Hölder inequality,
2π
Z
0
|f 0 (ρeiθ |p
dθ ≤
|f (ρeiθ )|p/2
2π
Z
0
|f 0 (ρeiθ )|2
dθ
|f (ρeiθ )|a
p/2 Z
2π
iθ
|f (ρe )|
bp
2−p
(2−p)/2
.
dθ
0
But
Z
0
2π
|f 0 (ρeiθ )|2
dθ
|f (ρeiθ )|a
p/2
Z
0
iθ
2
iθ
(2−a)−2
|f (ρe )| |f (ρe )|
=
p/2
2π
dθ
,
(305)
0
and we invoke the following Lemma, which is a consequence of Hardy’s identity and
Koebe’s theorem (see [57]):
Lemma 5.1. There exists a universal constant C > 0 such that, for f holomorphic
and injective in the unit disk, with f (0) = 0, f 0 (0) = 1,
(i) if p > 0,
Z
2π
|f 0 (ρeiθ )|2 |f (ρeiθ )|p−2 dθ ≤
0
C
;
(1 − ρ)2p+1
(ii) if p > 1/2,
Z
2π
|f (ρeiθ )|p dθ ≤
0
C
.
(1 − ρ)2p−1
Consider the last Lemma and (305); we seek for a such that 2 − a > 0 ⇔ b < 1,
together with
bp
1 1
> 1/2 ⇔ b > − ,
2−p
p 2
79
and we may find such a pair (a, b) iff p1 − 12 < 1 ⇔ p > 2/3.
With this condition on p satisfied, and for that choice of (a, b), Theorem 5.1 implies
that [note: 1 − r ≤ 1 − ρ ≤ 2(1 − r)],
Z
2π
|h0 (reiθ )|p dθ ≤ C(1 − r)−[2(2−a)+1)p/2] (1 − r)−[bp−(2−p)/2] ≤ C(1 − r)−(2p−1) .
0
The last statement shows that βh (p) ≤ 2p − 1 for 2/3 < p < 2. In order to prove it
for all p > 0, we first need the
Lemma 5.2. There exists a universal constant C > 0 such that
|h0 (z)| ≤ C(1 − r)−2 .
Proof: We write
s
0 (z 2 ) p
f
0 (z 2 ) ≤ C(1 − ρ)−1/2 (1 − ρ)−3/2 ,
|h0 (z)| = z
f
f (z 2 )
by the Koebe distortion theorem. (Use inequalities (11) and (13) on page 21 of [56].)
An immediate corollary of this Lemma is that βh (p) ≤ 2p if p > 0.
We now argue as in [57], using the fact that the function βh is convex. Any
number bigger than 2/3 may be written as p + q with 2/3 < p < 2 and q > 0. We
can then write
q
p
p + q = t + (1 − t)
t
1−t
where t ∈ [0, 1] is close to 1 and
βh (p + q) ≤ tβh
p
t
+ (1 − t)βh
q
1−t
p
≤ t 2 − 1 + 2q
t
by the Lemma above. Finally
βh (p + q) ≤ 2(p + q) − t
which converges to 2(p + q) − 1 as t → 1.
Acknowledgements
It is a pleasure to thank Michael Benedicks and Steffen Rohde for discussions, and
David Kosower for a reading of the manuscript. BD also wishes to thank the MSRI
at Berkeley for its kind hospitality during the Program “Random Spatial Processes”
(January 9, 2012 to May 18, 2012).
80
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