Ann. Henri Poincaré 4, Suppl. 1 (2003) S401 – S426
c Birkhäuser Verlag, Basel, 2003
1424-0637/03/01S401-26
DOI 10.1007/s00023-003-0931-0
Annales Henri Poincaré
Conformal Spiral Multifractals
Bertrand Duplantier
Abstract. The exact joint multifractal distribution for the scaling and spiraling
of electrostatic potential lines near any conformally invariant scaling curve is derived in two dimensions. Its spectrum f (α, λ) gives the Hausdorff dimension of
the points where the potential scales with distance r as H ∼ r α while the curve
spirals logarithmically with a rotation angle ϕ = λ ln r. It obeys the scaling law
f (α, λ) = (1 + λ2 )f (ᾱ) − bλ2 with ᾱ = α/(1 + λ2 ) and b = (25 − c)/12, and where
f (α) ≡ f (α, λ = 0) is the pure harmonic measure spectrum, and c the conformal
central charge. The results apply to O(N ) and Potts models, as well as to SLEκ .
1 Introduction
The geometric description of the random fractals arising in Nature is a fascinating
subject. Outstanding among these fractals is the class of random clusters or curves
arising in equilibrium critical phenomena, which are associated with fundamental
ideas of scale invariance. In particular, in two dimensions (2D), statistical systems
at their critical point are expected to produce conformally invariant (CI) fractal
structures [1], with a Gibbs equilibrium weight invariant under all planar conformal maps. This leads to a universal random geometry. Prominent examples are
the continuum scaling limits of random walks (RW), i.e., Brownian motion, selfavoiding walks (SAW), and critical percolation, Ising or Potts clusters. A wealth
of exact methods has been devised for their study: Coulomb gas [2], conformal
field theory (CFT) [3], and quantum gravity methods [4, 5, 6, 7, 8]. Recently, in
parallel to the “cascade structure” found by conformal invariance of Brownian
motion [9], a natural “multifractal structure” has been found from 2D quantum
gravity [10]. It was developed for self-avoiding walks [11] and percolation clusters
[12], (see also [13, 14, 15]), and finally for random curves with arbitrary central
charge c, the parameter labeling the universality class of the underlying CFT [16].
Rigorous probabilistic methods have also been developed, with the landmark introduction of the Stochastic Löwner Evolution (SLE) process, which directly mimics
the wandering of critical cluster boundaries in the scaling limit [17].
A refined way of accessing the random geometry of conformally invariant
scaling curves is provided by classical potential theory of electrostatic or diffusion
fields near these random fractal boundaries, whose self-similarity is reflected in
a multifractal (MF) spectrum describing the singularities of the potential, also
called harmonic measure. The first cases addressed were those of RW’s or SAW’s
in dimension d = 4−ε [18, 19]. In two dimensions, the first exact examples obtained
from the quantum gravity approach appeared for the universality class of random
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or self-avoiding walks [11] and percolation clusters [12], which all possess in 2D the
same harmonic MF spectrum (see also [9, 13, 14, 15]). The general solution for the
potential distribution near any CI fractal in 2D, obtained in [16], depends only on
the central charge (see also [20, 21]). This solution can be generalized to higher
multifractal correlations, like the joint distribution of potential on both sides of a
simple scaling path [22, 23].
On another hand, the study of the SLE process is presently a very active
field of research [24, 25, 26, 27, 28, 29, 30, 31] (see also the contribution [32] in
this volume). It is intimately related to the quantum gravity approach [33].
An important question concerns the geometry of the equipotential lines near
a random (CI) fractal curve. They are expected to rotate wildly, or wind, in a
spiraling motion, that closely follows the boundary itself. The key geometrical
object is here the logarithmic spiral, a structure which is conformally invariant.
The MF description should generalize to a mixed multifractal spectrum, accounting
for both scaling and winding of the equipotentials [34].
The exact solution to this mixed MF spectrum for any random CI curve has
been recently obtained in a collaboration with Ilia Binder in [35]. In particular, it
has been shown to be related by a scaling law to the usual harmonic MF spectrum.
A rigorous derivation from the SLE formalism is possible [36].
In these Proceedings, we describe the multifractal properties of the scaling
and spiraling of random curves, as seen by the harmonic measure. The universal
description is given in terms of the central charge c of the underlying conformal
field theory. The results apply to the critical O(N ) loop model, or to the EP’s
of critical Fortuin-Kasteleyn (FK) clusters in the Q-Potts model, all described in
terms of Coulomb gas with some coupling constant g [2]. SLEκ paths also describe
cluster frontiers or hulls. One has the correspondence κ = 4/g, with a central
charge
1
(1)
c = (3 − 2g)(3 − 2g ) = (6 − κ)(6 − κ ),
4
symmetric under the duality gg = 1 or κκ = 16 [16, 23, 33].
In the following sections, we present the universal description of multifractal
functions for arbitrary conformally-invariant curves. For several simple paths we
also define higher multifractal spectra, which include the spiraling rate. A more
complete description can be found in [33].
2 Conformally Invariant Frontiers and Quantum Gravity
2.1
Harmonic Measure and Potential near a Fractal Frontier
2.1.1 Local Behavior of the Potential
Consider a single (conformally-invariant) critical random cluster, generically called
C. Let H (z) be the potential at an exterior point z ∈ C, with Dirichlet boundary
conditions H (w ∈ ∂C) = 0 on the outer (simply connected) boundary ∂C of C,
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Conformal Spiral Multifractals
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(or frontier F := ∂C) , and H(w) = 1 on a circle “at ∞”, i.e., of a large radius
scaling like the average size R of C. As is well-known [37], H (z) is identical to
the harmonic measure of the circle “at ∞” seen from z, i.e, the probability that a
random walker (more precisely, a Brownian motion) launched from z, escapes to
∞ without having hit C.
The multifractal formalism [38, 39, 40, 41] characterizes subsets ∂C α of
boundary sites by a Hölder exponent α, and a Hausdorff dimension f (α) =
dim (∂C α ), such that their potential locally scales as
α
H (z → w ∈ ∂C α ) ≈ (|z − w|/R) ,
(2)
in the scaling limit a r = |z − w| R (with a the underlying lattice constant
if one starts from a lattice description before taking the scaling limit a → 0).
2.1.2 Equivalent Electrostatic Wedge Angle
In 2D the complex potential ϕ(z) (such that the electrostatic potential H(z) =
ϕ(z) and the field’s modulus |E(z)| = |ϕ (z)|) for a wedge of angle θ, centered at
w, is
(3)
ϕ(z) = (z − w)π/θ .
By eq. (2) a Hölder exponent α thus defines a local equivalent “electrostatic” angle
θ = π/α, and the MF dimension fˆ(θ) of the boundary subset with such θ is
fˆ(θ) = f (α = π/θ).
(4)
2.1.3 Moments
It is convenient to define the harmonic measure H(w, r) = H(∂C ∩ B(w, r)) in a
ball B(w, r) of radius r centered ar w ∈ ∂C, as the probability that a Brownian
path started at infinity first hits the frontier F = ∂C inside the ball B(w, r). It is
the integral of the Laplacian of potential H in the ball B(w, r), i.e., the boundary
charge in that ball. It thus scales as rα with the same exponent as in (2).
Of special interest are the moments of H, averaged over all realizations of C,
and defined as
n
Zn =
H (w, r) ,
(5)
z∈∂C/r
where points w ∈ ∂C/r are the centers of a covering of the frontier ∂C by balls of
radius r, and form a discrete subset ∂C/r ⊂ ∂C. The moment order n can be a
real number. In the scaling limit, one expects these moments to scale as
Zn ≈ (r/R)
τ (n)
,
(6)
where the multifractal scaling exponents τ (n) vary in a non-linear way with n
[38, 39, 40, 41]. They obey the symmetric Legendre transform τ (n) + f (α) = αn,
with n = f (α) , α = τ (n). By normalization: τ (1) = 0. Because of the ensemble
average (5), values of f (α) can become negative for some domains of α [18, 42].
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Bertrand Duplantier
Ann. Henri Poincaré
Calculation of Exponents from Quantum Gravity
2.2.1 Brownian Representation of Moments
A collection of n independent RW’s, or Brownian paths B in the scaling limit,
started at the same point a distance r away from the cluster’s hull’s frontier
∂C, and diffusing without hitting ∂C, give a geometric representation of the nth
moment, H n , in eq.(5) for n integer. Convexity yields the analytic continuation to
arbitrary n’s. Let us introduce the notation A ∧ B for two random sets required to
traverse, without mutual intersection, the annulus D (r, R) from the inner boundary
circle of radius r to the outer one at distance R, and A ∨ B for two independent,
thus possibly intersecting, sets [11]. With this notation, one can define a grand
canonical partition function which describes the star configuration of the Brownn
ian paths and cluster: ∂C ∧ n := ∂C ∧ (∨B) . At the critical point, it is expected
to scale for r/R → 0 as
x(n)+···
ZR (∂C ∧ n) ≈ (r/R)
,
(7)
where the scaling exponent
x (n) := x (∂C ∧ n)
(8)
depends on n and is associated with the conformal operator creating the star
vertex ∂C ∧ n. The dots after exponent x(n) express the fact that there may be an
additional contribution to the exponent, independent of n, corresponding to the
entropy associated with the extremities of the random frontier [11].
By normalization, this contribution actually does not appear in the multifractal moments. Since H is a probability measure, the sum (5) is indeed normalized
as
Zn=1 = 1,
(9)
or in terms of star partition functions:
Zn = ZR (∂C ∧ n) /ZR (∂C ∧ 1) .
(10)
The scaling behavior (7) thus gives
Zn ≈ (r/R)x(n)−x(1) .
(11)
The last exponent actually obeys the identity x(1) = x (∂C ∧ 1) = 2, which will
be obtained directly, and can also be seen as a consequence of Gauss’s theorem in
two dimensions [18].
Owing to eqs. (6) (11) we simply get
τ (n) = x(n) − x (1) = x (n) − 2.
(12)
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2.2.2 Quantum Gravity
To calculate these exponents, we use the fundamental mapping between the conformal field theory, describing a critical statistical system in the plane C or half-plane
H, and the same CFT on a random planar surface, i.e., in presence of quantum
gravity (QG) [5, 6]. Two universal functions U and V , which now depend on the
central charge c of the CFT, describe the KPZ map between conformal dimensions
∆ in bulk or boundary QG and those in the standard plane or half-plane:
U (∆) := ∆
1 ∆2 − γ 2
∆−γ
, V (∆) =
,
1−γ
4 1−γ
with
V (∆) := U
1
(∆ + γ) .
2
(13)
(14)
The parameter γ is the string susceptibility exponent of the random 2D surface (of
genus zero), bearing the CFT of central charge c [5]; γ is the solution of
c = 1 − 6γ 2 (1 − γ)−1 , γ ≤ 0.
(15)
The function U maps quantum gravity conformal weights, whether in the bulk
˜ into their counterparts in C, x = 2U (∆), or in H,
(∆) or on a boundary (∆),
˜
x̃ = U (∆). The function V has been tailored to map quantum gravity boundary
˜ to the corresponding conformal dimensions x = 2V (∆)
˜ in the full
dimensions ∆
plane C.
The positive inverse function of U , U −1 , is
1 4(1 − γ)x + γ 2 + γ ,
(16)
U −1 (x) =
2
and transforms conformal weights of a conformal operator in C or H into the
conformal weights of the same operator in quantum gravity, in the bulk or on the
boundary.
2.2.3 Boundary Additivity Rules
Consider two arbitrary random sets A, B, with boundary scaling exponents x̃ (A) ,
x̃ (B) in the half-plane H with Dirichlet boundary conditions. When these two sets
are mutually-avoiding, the scaling exponent x (A ∧ B) in C, as in (8), or x̃ (A ∧ B)
in H have the universal structure [11, 12, 16]
(17)
x (A ∧ B) = 2V U −1 (x̃ (A)) + U −1 (x̃ (B)) ,
x̃ (A ∧ B) =
U U −1 (x̃ (A)) + U −1 (x̃ (B)) .
(18)
These fundamental relations generalize the “cascade relations” found in [9] for
Brownian paths. They are established in the general case in [33]. U −1 (x̃) is, on the
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1
1
U (~
x 2) = 2 U (~
x 1)
Figure 1: Illustration of the additivity rule (20): each of the two non-intersecting
strands of a simple random path lives in its own sector of the random disk near
the Dirichlet boundary.
random disk with Dirichlet boundary conditions, the boundary scaling dimension
corresponding to x̃ in the half-plane H, and in eqs. (17) (18)
U −1 (x̃ (A ∧ B)) =
U −1 (x̃ (A)) + U −1 (x̃ (B))
(19)
is a linear boundary exponent corresponding to the fusion of two “boundary operators” on the random disk, under the Dirichlet mutual avoidance condition A ∧ B.
This quantum boundary conformal dimension is mapped back by V to the scaling
dimension in C, or by U to the boundary scaling dimension in H [16] (see [33]).
2.2.4 Exponent Construction
For determining the harmonic exponents x(n) (8), we use (17) for A = ∂C and
n
B = (∨B) .
• We first need the boundary (conformal) scaling dimension (b.s.d.) x̃2 := x̃ (∂C)
associated with the presence of the random frontier near the Dirichlet boundary
H. Since this frontier is simple, it can be seen as made of two non-intersecting
semi-infinite strands (Fig. 1). Its b.s.d. in quantum gravity thus obeys (19)
U −1 (x˜2 ) = 2U −1 (x˜1 ) ,
(20)
where x̃1 is the boundary scaling dimension of a semi-infinite frontier path originating at the boundary H.
n
• The packet of n independent Brownian paths has x̃ ((∨B) ) = n, since x̃ (B) = 1.
• From (19) the QG boundary dimension of the whole set is (see Fig. 2):
˜ := U −1 [x̃ (∂C ∧ n)] = 2U −1 (x˜1 ) + U −1 (n) .
∆
(21)
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~
x2
n
1
~
x1
~
x1
1
1
U (~
x 1)
x 2) = 2U ( ~
U ( n)
1
1
~
x 1)
∆ = U ( n) + 2U ( ~
x (n)= 2U(∆)
1 ~
∆ = (∆ + γ )
2
Figure 2: The quantum gravity construction (20) (21) of exponents (22).
˜ + γ). From eqs.
Its associated QG bulk conformal dimension is therefore ∆ = 12 (∆
(14) or (17) we finally find
˜
x (n) = 2U (∆) = 2V (∆)
−1
= 2V 2U (x˜1 ) + U −1 (n) .
(22)
The whole construction is illustrated in Fig. 2.
• The value of the QG b.s.d. of a simple semi-infinite random path is
U −1 (x˜1 ) =
1
(1 − γ).
2
(23)
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Ann. Henri Poincaré
It is derived in [33] from the exponents of the O(N ) model, or of the SLE. It can
be directly derived from Makarov’s theorem:
α(n = 1) = τ (n = 1) =
dx
(n = 1) = 1,
dn
(24)
which, applied to (22), leads to the same result. We thus finally get
x (n) = 2V 1 − γ + U −1 (n) = 2U
1 1 −1
+ U (n) .
2 2
(25)
This result satisfies the identity: x(1) = 2U (1) = 2, which is related to Gauss’s
theorem, as mentioned above.
2.2.5 Multifractal Exponents
• The multifractal exponents τ (n) (12) are obtained from (13-16) as [16]
τ (n) =
=
x(n) − 2
1
12−γ (n − 1) +
[ 4(1 − γ)n + γ 2 − (2 − γ)].
2
41−γ
(26)
Similar exponents, but associated with moments taken at the tip, later appeared
in the context of the SLE process (see II in [43], and [44]; see also [21] for Laplacian
random walks, and [15] for SAW’s.) More multifractal exponents can be found for
the SLE in [33].
It is convenient to express the results in terms of the central charge c as:
• Multifractal Exponents
25 − c
24n + 1 − c
1
(n − 1) +
−1 ,
(27)
τ (n) =
2
24
25 − c
1−c
, +∞ ;
n ∈
n∗ = −
24
• Multifractal Spectrum
α
f (α)
α
dτ
1 1
25 − c
=
(n) = +
;
dn
2 2 24n + 1 − c
25 − c
1
1−c
=
α,
3−
−
48
2α − 1
24
1
, +∞ .
∈
2
(28)
(29)
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2.3
Conformal Spiral Multifractals
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Geometrical Analysis of Multifractal Spectra
2.3.1 Makarov’s Theorem
The exponents τ (n) satisfy, for any c, τ (n = 1) = 1, or equivalently f (α = 1) = 1,
i.e., Makarov’s theorem [45], valid for any simply connected boundary curve.
2.3.2 Symmetries
It is interesting to note that the general multifractal function (29) can also be
written as
25 − c
1
1
f (α) − α =
1−
2α − 1 +
.
(30)
24
2
2α − 1
The multifractal functions f (α)−α = fˆ(θ)− πθ thus possess the invariance property
upon substitution of primed variables given by
π
π
2π = θ + θ = + ;
(31)
α α
this corresponds to the complementary domain of the wedge θ. This basic symmetry, first observed [46] for the c = 0 result of [11], is valid for any conformally
invariant boundary.
2.3.3 Equivalent Wedge Distribution
The geometrical multifractal distribution of wedges θ along the boundary takes
the form:
π π
25 − c (π − θ)2
.
(32)
= −
fˆ(θ) = f
θ
θ
12 θ(2π − θ)
Remarkably enough, the second term also describes the contribution by a wedge
to the density of electromagnetic modes in a cavity [47]. The simple shift in (32),
25 → 25 − c, from the c = 0 case to general values of c, can then be related [48, 23]
to results of conformal invariance in a wedge [49].
2.3.4 Hausdorff Dimension of the External Perimeter
The maximum of f (α) corresponds to n = 0, and gives the Hausdorff dimension
DEP of the support of the measure, i.e., the accessible or external perimeter, a
simple curve without double points which may differ from the full hull [50, 13], as:
DEP
=
=
supα f (α) = f (α(n = 0))
√
√
1√
3
−
1 − c 25 − c − 1 − c .
2 24
(33)
(34)
This corresponds to a typical singularity exponent α̂ = α(0) = (3 − 2DEP )−1 , and
to a typical wedge angle θ̂ = π/α̂ = π(3 − 2DEP).
Bertrand Duplantier
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S410
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c=1
c=1/2
1.0
c=−2
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Ann. Henri Poincaré
0
5
1/2
10
α
c=0
15
20
Figure 3: Universal harmonic multifractal spectra f (α). The curves are indexed by
the central charge c, and correspond to: 2D spanning trees (c = −2); self-avoiding
or random walks, and percolation (c = 0); Ising clusters or Q = 2 Potts clusters
(c = 12 ); N = 2 loops, or Q = 4 Potts clusters (c = 1). The maximal dimensions are
those of the accessible frontiers. The left branches of the various f (α) curves are
largely indistinguishable, while their right branches split for large α, corresponding
to negative values of n.
2.3.5 Universal Multifractal Data
In Figure 3 we display the multifractal functions f , eq. (29), corresponding to
various values of −2 ≤ c ≤ 1, or, equivalently, to a number of components N ∈
[0, 2], and Q ∈ [0, 4] in the O(N ) or Potts models. The f (α) functions distinguish
the various universality classes, especially for negative n, i.e. large α.
2.3.6 Needles
The singularity at α = 12 , or θ = 2π, in the multifractal functions f , or fˆ, corresponds to boundary points with a needle local geometry, and Beurling’s theorem
[51] indeed insures that the Hölder exponents α are bounded below by 12 . This
corresponds to large values of n.
2.3.7 Fjords
The right branch of f (α) has a linear asymptote
lim f (α) /α = n∗ = −(1 − c)/24.
α→∞
(35)
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The α → ∞ behavior corresponds to moments of lowest order n → n∗ . This
describes almost inaccessible sites: Define N (H) as the number of boundary sites
having a given probability H to be hit by a RW starting at infinity; the MF
formalism yields, for H → 0, a power law behavior
∗
N (H) |H→0 ≈ H −(1+n
with an exponent 1 + n∗ =
23+c
24
)
(36)
≤ 1.
2.3.8 Ising Clusters
A critical Ising cluster (c = 12 ) possesses a multifractal spectrum with respect to
the harmonic measure:
1
7 √
τ (n) =
(n − 1) +
48n + 1 − 7 ,
(37)
2 48 1
49
1
α
f (α) =
, +∞ ,
(38)
3−
− , α∈
96
2α − 1
48
2
with the dimension of the accessible perimeter DEP = supα f (α, c = 12 ) =
11
8 .
2.3.9 Q = 4 Potts Clusters, and “ultimate Norway”
The limit multifractal spectrum is obtained for c = 1, which is an upper or lower
bound for all c’s, depending on the position of α with respect to 1:
f (α, c < 1) < f (α, c = 1), 1 < α,
f (α = 1, c) = 1, ∀c,
f (α, c < 1) > f (α, c = 1), α < 1.
This MF spectrum provides an exact example of a left-sided MF spectrum, with an
asymptote f (α → ∞, c = 1) = 32 (Fig. 3). It corresponds to singular boundaries
where fˆ (θ → 0, c = 1) = 32 = DEP , i.e., where the external perimeter is everywhere
dominated by “fjords”, with typical angle θ̂ = 0. It is tempting to call it the
“ultimate Norway”.
The frontier of a Q = 4 Potts Fortuin-Kasteleyn cluster, or the SLEκ=4
provide such an example for this left-handed multifractal spectrum (c = 1). The
MF data are:
√
1
τ (n) =
(n − 1) + n − 1,
(39)
2
1
1
1
, +∞ ,
(40)
f (α) =
3−
, α∈
2
2α − 1
2
with accessible sites forming a set of Hausdorff dimension
DEP = supα f (α, c = 1) =
3
.
2
(41)
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The external perimeter which bears the electrostatic charge is a non-intersecting
simple path. We therefore arrive at the striking conclusion that in the plane, a
conformally-invariant scaling curve which is simple has a Hausdorff dimension at
most equal to DEP = 3/2 [16]. The corresponding Q = 4 Potts frontier, while still
possessing a set of double points of dimension 0, actually develops a logarithmically
growing number of double points [52].
2.3.10 Dimensions of Potts Hulls and Frontiers
As an application of the above results, let us give the various dimensions appearing
along Potts cluster boundaries. These dimensions have been recently beautifully
checked numerically [53].
Q
c
DEP
DH
DSC
0
-2
5/4
2
5/4
1
0
4/3
7/4
3/4
2
1/2
11/8
5/3
13/24
3
4/5
17/12
8/5
7/20
4
1
3/2
3/2
0
Table 1: Dimensions for the critical Q-state Potts model; Q = 0, 1, 2 correspond
to spanning trees, percolation and Ising clusters, respectively.
3 Higher Universal Multifractal Spectra
3.1
Double-Sided Spectrum: Definition
We consider here the specific case where the fractal set C is a (conformallyinvariant) simple scaling curve, that is, it does not contain double points. The
frontier ∂C is thus identical with the set itself: ∂C = C. Each point of the curve can
then be reached from infinity, and we address the question of the simultaneous
behavior of the potential on both sides of the curve. Notice, however, that one
can also address the case of non-simple random paths, by concentrating on the
double-sided potential near cut-points, as for instance in Brownian paths [33].
The potential H scales as
(42)
H+ z → w+ ∈ ∂C α,α ≈ |z − w|α ,
when approaching w on one side of the scaling curve, while scaling as
H− z → w− ∈ ∂C α,α ≈ |z − w|α ,
(43)
on the other side. The multifractal formalism now characterizes subsets Cα,α of
boundary sites w with two such Hölder exponents, α, α , by their Hausdorff dimension f2 (α, α ) := dim (Cα,α ). The standard one-sided multifractal spectrum
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f (α) is then recovered as the supremum:
f (α) = supα f2 (α, α ) .
3.2
(44)
Explicit Higher Spectra
3.2.1 Multiple-Sided Spectra
One can consider a star configuration Sm of a number m of similar simple scaling paths, all originating at the same vertex w, and the joint harmonic measure
moments in each branch. These higher multifractal spectra require some further
interpretation as associated with the scaling behavior of densities of points, as they
yield negative dimensions for high enough m [23].
Their evaluation for an m-arm star gives the explicit formula in terms of the
central charge c [23]:
fm ({αi=1,...,m }) =
−1
m
1
25 − c
1 −1
2
−
m 1−
α
12
8(1 − γ)
2 i=1 i
m
−
1−c αi ,
24 i=1
(45)
where the central charge c and the parameter γ are related by eq. (15). The domain
of definition of the poly-multifractal function f is independent of c and given by
m
1−
1 −1
α ≥ 0.
2 i=1 i
(46)
3.2.2 One and Two-Sided Cases
The m = 1 case
f1 (α) =
1
25 − c
−
12
8(1 − γ)
1
1−c
α
1−
−
2α
24
(47)
corresponds to the potential in the vicinity of the tip of a conformally-invariant
scaling path, and naturally differs from the usual f (α) spectrum, which describes
the potential on one side of the scaling path.
The two-sided case mentioned in section 3.1 is obtained for m = 2 as
−1
1
1
25 − c
1 1
−
+
f2 (α, α ) =
1−
12
2(1 − γ)
2 α α
1−c
(α + α ) .
−
(48)
24
This doubly multifractal spectrum possesses the desired properties, like (44).
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Bertrand Duplantier
Ann. Henri Poincaré
4 Spiraling of Conformally Invariant Curves
Another important question arises concerning the geometry of the equipotential
lines near a random (CI) fractal curve. These lines are expected to rotate wildly,
or wind, in a spiraling motion that closely follows the boundary itself. The key
geometrical object is here the logarithmic spiral, which is conformally invariant
(Fig. 4). The MF description should generalize to a mixed multifractal spectrum,
2.5
2
1.5
1
0.5
0
−0.5
−1
−1.5
−2
−3
−2
−1
0
1
2
3
Figure 4: A double logarithmic spiral mimicking the local geometry of the two
strands of the conformally-invariant frontier.
accounting for both scaling and winding of the equipotentials [34].
In this section, we describe the exact solution to this mixed MF spectrum for
any random CI curve [35]. In particular, it is shown to be related by a scaling law
to the usual harmonic MF spectrum. We use the same conformal tools as before,
fusing quantum gravity and Coulomb gas methods, which allow the description
of Brownian paths interacting and winding with CI curves, thereby providing a
probabilistic description of the potential map near any CI random curve.
4.1
Harmonic Measure and Rotations
As above, consider a single critical random cluster C in the scaling limit. Let H (z)
be the potential at an exterior point z ∈ C, with Dirichlet boundary conditions
H (w ∈ ∂C) = 0 on the outer boundary ∂C of C, and H(w) = 1 on a circle “at ∞”,
i.e., of a large radius scaling like the average size R of C.
Let us now consider the degree with which the external boundary curve (which
is simply connected and conformally invariant) winds in the complex plane about
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Conformal Spiral Multifractals
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the boundary point w ∈ ∂C. Let us generically call ϕ(z) = arg (z −w). In the scaling
limit, the multifractal formalism, here generalized to take into account rotations
[34], characterizes subsets ∂C α,λ of boundary sites by a Hölder exponent α, and a
rotation rate λ, such that their potential lines respectively scale and logarithmically
spiral as
H (z → w ∈ ∂C α,λ ) ≈
rα ,
ϕ (z → w ∈ ∂C α,λ ) ≈
λ ln r ,
(49)
in the limit r = |z − w| → 0. The Hausdorff dimension dim (∂C α,λ ) = f (α, λ)
defines the mixed MF spectrum, which is CI since under a conformal map both α
and λ are locally invariant.
As above, we consider the harmonic measure H (w, r), which is the integral
of the Laplacian of H in a disk B(w, r) of radius r centered at w ∈ ∂C, i.e., the
boundary charge in that disk. It scales as rα with the same exponent as in (49),
and is also the probability that a Brownian path started at large distance R first
hits the boundary at a point inside B(w, r). Let ϕ(w, r) be the associated winding
angle of the path down to distance r from w. The mixed moments of H and eϕ ,
averaged over all realizations of C, are defined as
τ (n,p)
n
Zn,p =
H (w, r) exp (p ϕ(w, r)) ≈ (r/R)
,
(50)
w∈∂C/r
where the sum runs over the centers of a covering of the boundary by disks of
radius r, and where n and p are real numbers. As before, the nth moment of
H (w, r) is the probability that n independent Brownian paths diffuse along the
boundary and all first hit it at points inside the disk B(w, r). The angle ϕ(w, r) is
then their common winding angle down to distance r (Fig. 5).
The scaling limit in (50) involves multifractal scaling exponents τ (n, p) which
vary in a non-linear way with n and p. They give the multifractal spectrum f (α, λ)
via a symmetric double Legendre transform:
α
f (α, λ)
n
=
∂τ
(n, p) ,
∂n
λ=
∂τ
(n, p) ,
∂p
= αn + λp − τ (n, p) ,
=
∂f
(α, λ) ,
∂α
p=
∂f
(α, λ) .
∂λ
(51)
Because of the ensemble average (50), f (α, λ) can become negative for some α, λ.
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Bertrand Duplantier
R
Ann. Henri Poincaré
r
Figure 5: Two-sided boundary curve ∂C and Brownian n-packet winding together
from the disk of radius r up to distances of order R, as measured by the winding
angle ϕ(w, r) = arg(∂C ∧ n) as in (50) and in (58).
4.2
Exact Mixed Multifractal Spectrum
The 2D conformally invariant random statistical system is labelled by its central
charge c, c ≤ 1 [1]. The main result is the following exact scaling law [35]:
α
2
f (α, λ) = (1 + λ )f
(52)
− bλ2 ,
1 + λ2
25 − c
≥2,
b :=
12
where f (α) = f (α, λ = 0) is the usual harmonic MF spectrum in the absence of
prescribed winding, first obtained in [16], and described in section 2, eq. (29). It
can be recast as:
f (α)
b
= α+b−
=
25 − c
.
12
bα2
,
2α − 1
(53)
We thus arrive at the very simple formula for the mixed spectrum:
f (α, λ) = α + b −
bα2
.
2α − 1 − λ2
Notice that by conformal symmetry
supλ f (α, λ) = f (α, λ = 0),
(54)
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Conformal Spiral Multifractals
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i.e., the most likely situation in the absence of prescribed rotation is the same as
λ = 0, i.e. winding-free. The domain of definition of the usual f (α) (54) is 1/2 ≤ α
[16, 51], thus for λ-spiraling points eq. (52) gives
1
(1 + λ2 ) ≤ α,
2
(55)
in agreement with a theorem by Beurling [51, 34].
We have seen in section 2.3 the geometrical meaning to the exponent α:
For an angle with opening θ, α = π/θ, the quantity π/α can be regarded as a
local generalized angle with respect to the harmonic measure. The geometrical
MF spectrum of the boundary subset with such opening angle θ and spiraling rate
λ reads from (54)
π
π 1
1
π
ˆ
+ 2π
.
f (θ, λ) := f (α = , λ) = + b − b
θ
θ
2 θ
1+λ2 − θ
As in (55), the domain of definition in the θ variable is
0 ≤ θ ≤ θ(λ),
θ(λ) = 2π/(1 + λ2 ).
The maximum is reached when the two frontier strands about point w locally
collapse into a single λ-spiral, whose inner opening angle is θ(λ) [51].
In the absence of prescribed winding (λ = 0), the maximum DEP :=
DEP (0) = supα f (α, λ = 0) gives the dimension of the external perimeter of the
fractal cluster. Its dimension (34) reads in this notation
DEP =
1
1
25 − c
(1 + b) −
.
b(b − 2), b =
2
2
12
For spirals, the maximum value DEP (λ) = supα f (α, λ) still corresponds in
the Legendre transform (51) to n = 0, and gives the dimension of the subset of
the external perimeter made of logarithmic spirals of type λ. Owing to (52) we
immediately get
DEP (λ) = (1 + λ2 )DEP − bλ2 .
(56)
This corresponds to typical scaled values
α̂(λ) = (1 + λ2 )α̂, θ̂(λ) = θ̂/(1 + λ2 ).
Since b ≥ 2 and DEP ≤ 3/2, the EP dimension decreases with spiraling rate, in a
simple parabolic way.
Fig. 6 displays typical multifractal functions f (α, λ; c). The example chosen, c = 0, corresponds to the cases of a SAW, or of a percolation EP, the scaling limits of which both coincide with the Brownian frontier [11, 12, 14]. The
original singularity at α = 12 in the rotation free MF functions f (α, 0), which
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Bertrand Duplantier
Ann. Henri Poincaré
f(α,λ)
1.5
f(3,0)=4/3
1.0
0.5
λ=0.
λ=.7
λ=1.
0.0
0
1/2
5
10
α
15
20
Figure 6: Universal multifractal spectrum f (α, λ) for c = 0 (Brownian frontier,
percolation EP and SAW), and for three different values of the spiraling rate λ.
The maximum f (3, 0) = 4/3 is the Hausdorff dimension of the frontier.
describes boundary points with a needle local geometry, is shifted for λ = 0 towards the minimal value (55). The right branch of f (α, λ) has a linear asymptote
limα→+∞ f (α, λ) /α = −(1 − c)/24. Thus the λ-curves all become parallel for
α → +∞, i.e., θ → 0+ , corresponding to deep fjords where winding is easiest.
Limit multifractal spectra are obtained for c = 1, which exhibit exact examples of left-sided MF spectra, with a horizontal asymptote f (α → +∞, λ; c = 1) =
3
1 2
2 − 2 λ (Fig. 7). This corresponds to the frontier of a Q = 4 Potts cluster (i.e., the
SLEκ=4 ), a universal random scaling curve, with the maximum value DEP = 3/2,
and a vanishing typical opening angle θ̂ = 0, i.e., the “ultimate Norway” where
the EP is dominated by “fjords” everywhere [16, 23]. Fig. 8 displays the dimension
DEP (λ) as a function of the rotation rate λ, for various values of c ≤ 1, corresponding to different statistical systems. Again, the c = 1 case shows the least
decay with λ, as expected from the predominance of fjords there.
4.3
Conformal Invariance and Quantum Gravity
We now give the main lines of the derivation of exponents τ (n, p), hence f (α, λ)
[35]. As before, n independent Brownian paths B, starting a small distance r away
from a point w on the frontier ∂C, and diffusing without hitting ∂C, give a geometric
representation of the nth moment, H n , of the harmonic measure in eq.(50) for
integer n (Fig. 5). Convexity yields the analytic continuation to arbitrary n’s.
Let us introduce an abstract (conformal) field operator Φ∂C∧n characterizing the
presence of a vertex where n such Brownian paths and the cluster’s frontier diffuse
away from each other in the mutually-avoiding configuration ∂C ∧n [11, 12]; to this
operator is associated a scaling dimension x(n). To measure rotations using the
moments (50) we have to consider expectation values with insertion of the mixed
Vol. 4, 2003
Conformal Spiral Multifractals
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f(α,λ)
1.5
1.0
0.5
λ=0.
λ=.7
λ=1.
0.0
0
1/2
5
10
α
15
20
Figure 7: Left-sided multifractal spectra f (α, λ) for the limit case c = 1, the
“ultimate Norway” (frontier of a Q = 4 Potts cluster or SLEκ=4 ).
operator
Φ∂C∧n ep arg(∂C∧n) −→ x (n, p) ,
(57)
where arg(∂C ∧n) is the winding angle common to the frontier and to the Brownian
paths (see Fig. (5)), and where x(n, p) is the scaling dimension of the operator
Φ∂C∧n ep arg(∂C∧n) . It is directly related to τ (n, p) by [11]
x (n, p) = τ (n, p) + 2.
(58)
For n = 0, one recovers the previous scaling dimension
x(n, p = 0) =
x(n),
τ (n, p = 0) =
τ (n) = x (n) − 2.
For the purely harmonic exponents x(n), describing the mutually-avoiding set
∂C ∧ n, we have seen in eqs. (25) and (20) that
(59)
x(n) = 2V 2U −1 (x˜1 ) + U −1 (n) .
In (59), we recall that the arguments x˜1 and n are respectively the boundary
scaling dimensions (b.s.d.) (20) of the simple path S1 representing a semi-infinite
random frontier (such that ∂C = S1 ∧ S1 ), and of the packet of n Brownian paths,
both diffusing into the upper half-plane H. The function U −1 transforms these
half-plane b.s.d’s into the corresponding b.s.d.’s in quantum gravity, the linear
combination of which gives, still in QG, the b.s.d. of the mutually-avoiding set
∂C ∧ n = (∧S1 )2 ∧ n. The function V finally maps the latter b.s.d. into the scaling
dimension in C. The path b.s.d. x˜1 (20) obeys U −1 (x˜1 ) = (1 − γ)/2.
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Bertrand Duplantier
Ann. Henri Poincaré
2
DEP(λ)
1
c=1
c=1/2
c=0
c=−2
0
−1
−2
−3.0
−2.0
−1.0
0.0
λ
1.0
2.0
3.0
Figure 8: Dimensions DEP (λ) of the external frontiers as a function of rotation
rate. The curves are indexed by the central charge c, and correspond respectively
to: loop-erased RW (c = −2; SLE2 ); Brownian or percolation external frontiers,
and self-avoiding walk (c = 0; SLE8/3 ); Ising clusters (c = 12 ; SLE3 ); Q = 4 Potts
clusters (c = 1; SLE4 ).
It is now useful to consider k semi-infinite random paths S1 , joined at a single
k
vertex in a mutually-avoiding star configuration Sk =S1 ∧ S1 ∧ · · · S1 = (∧S1 )k . (In
this notation the frontier near any of its points is a two-star ∂C = S2 .) The scaling
dimension of Sk can be obtained from the same b.s.d. additivity rule in quantum
gravity, as in (17) or (59) [16]
x(Sk )
= 2V k U −1 (x˜1 ) .
(60)
The scaling dimensions (59) and (60) coincide when
x(n)
=
x(Sk(n) )
(61)
−1
k(n) =
2+
U (n)
.
U −1 (x˜1 )
(62)
Thus we state the scaling star-equivalence
∂C ∧ n ⇐⇒ Sk(n) ,
(63)
of two mutually-avoiding simple paths ∂C = S2 = S1 ∧ S1 , further avoiding n
Brownian motions, to k(n) simple paths in a mutually-avoiding star configuration
Sk(n) (Fig. 9). This equivalence plays an essential role in the computation of the
complete rotation spectrum (58).
Vol. 4, 2003
Conformal Spiral Multifractals
S421
1
2
n
k (n) = 2 + U ( )
1
x)
U (~
n
1
Figure 9: Equivalence (62) between two simple paths in a mutually-avoiding configuration S2 = S1 ∧ S1 , further avoided by a packet of n independent Brownian
motions, and k(n) simple paths in a mutually-avoiding star configuration Sk(n) .
4.4
Rotation Scaling Exponents
The Gaussian distribution of the winding angle about the extremity of a scaling
path, like S1 , was derived in [54], using exact Coulomb gas methods. The argument
can be generalized to the winding angle of a star Sk about its center [55], where
one finds that the angular variance is reduced by a factor 1/k 2 (see also [56]).
The scaling dimension associated with the rotation scaling operator ΦSk ep arg(Sk )
is found by analytic continuation of the Fourier transforms evaluated there [35]:
x(Sk ; p) = x(Sk ) −
2 p2
,
1 − γ k2
i.e., is given by a quadratic shift in the star scaling exponent. To calculate the
scaling dimension (58), it is sufficient to use the star-equivalence (62) above to
conclude that
x(n, p) = x(Sk(n) ; p) = x(n) −
p2
2
,
2
1 − γ k (n)
which is the key to our problem. Using eqs. (62), and (59) gives the useful identity:
1
(1 − γ)k 2 (n) = x(n) − 2 + b,
8
with b =
2
1 (2−γ)
2 1−γ
=
25−c
12 .
Recalling (58), we arrive at the multifractal result:
τ (n, p) = τ (n) −
p2
1
,
4 τ (n) + b
(64)
where τ (n) = x(n) − 2 corresponds to the purely harmonic spectrum with no
prescribed rotation.
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4.5
Bertrand Duplantier
Ann. Henri Poincaré
Legendre Transform
The structure of the full τ -function (64) leads by a formal Legendre transform (51)
directly to the identity
f (α, λ)
=
(1 + λ2 )f (ᾱ) − bλ2 ,
where f (ᾱ) ≡ ᾱn − τ (n), with ᾱ = dτ (n)/dn, is the purely harmonic
MF function.
It depends on the natural reduced variable ᾱ à la Beurling (ᾱ ∈ 12 , +∞ )
1 1
b
dx
α
(n) = +
,
=
ᾱ :=
1 + λ2
dn
2 2 2n + b − 2
whose expression emerges explicitly from (59). Whence eq.(52), QED.
4.6
Higher Mixed Rotation Spectra
It is interesting to consider also mixed higher multifractal spectra including logarithmic spiraling rates [57]. For a conformally-invariant scaling curve which is
simple, i.e., without double points, like the external frontier ∂C, here taken alone,
define the universal function f2 (α, α ; λ) which gives the Hausdorff dimension of
the points where the potential varies jointly with distance r as rα on one side of
the curve, and as rα on the other, given a local spiraling rate λ of the curve. It
satisfies the generalization of scaling relation (52) in terms of the scaled variables
ᾱ = α/(1 + λ2 ), ᾱ = α /(1 + λ2 )
f2 (α, α ; λ)
=
=
(1 + λ2 )f2 (ᾱ, ᾱ ; λ = 0) − bλ2 .
α
α
,
(1 + λ2 )f2
− bλ2 ,
1 + λ2 1 + λ2
(65)
(66)
where
f2 (α, α ) = f2 (α, α ; λ = 0) =
−1
1
1
1
− 1−
2(1 − γ)
2α 2α
b−2
(α + α )
−
2
b−
(67)
is the standard double-sided multifractal distribution (48) for the potential. The
mixed double-sided multifractal function is thus
−1
1
1
1
1
−
f2 (α, α ; λ) = b −
−
2(1 − γ) 1 + λ2
2α 2α
b−2
(α + α ) .
−
(68)
2
This double multifractality can be generalized to higher ones [23], by considering
the distribution of potential between the arms of a rotating star Sm . It can be
Vol. 4, 2003
Conformal Spiral Multifractals
S423
expressed via the scaled variables
ᾱi =
αi
,
1 + λ2
as
fm ({αi }; λ) = (1 + λ2 )fm ({ᾱi }; λ) − bλ2
αi
= (1 + λ2 )fm
− bλ2 ,
1 + λ2
(69)
(70)
in terms of the purely harmonic multifractal spectrum (45):
fm ({αi }) =
=
fm ({αi }; λ = 0)
−1
m
1
1
2
m 1−
b−
8(1 − γ)
2αi
i=1
(71)
m
−
b−2
αi .
2 i=1
This gives the following poly-multifractal result [57]:
−1
m
1
1
1
2
m
fm ({αi }; λ) = b −
−
8(1 − γ)
1 + λ2 i=1 2αi
m
−
b−2
αi .
2 i=1
(72)
In conclusion, we have described general conformally-invariant curves in the
plane in terms of the universal parameters c (central charge) or γ (string susceptibility). The multifractal results described in the sections above thus also apply
to the SLEκ process after substituting κ for γ or c (Eq. (1). Two dual values of
κ correspond to the same values of γ or c. The reason is that we have considered
geometrical properties of the boundaries which actually were self-dual. An exemple
is the harmonic multifractal spectrum of the SLEκ ≥4 frontier, which is identical
to that of the smoother (simple) SLE(16/κ )=κ≤4 path. So we actually need only
the set of simple SLE traces with κ ≤ 4. The mixed harmonic spectrum f (α, λ),
which can also be derived directly for the SLE [36], obeys the same duality property. When dealing with higher multifractality, we assumed the random curves to
be simple, thus again κ ≤ 4or non simple random paths, quantum gravity rules
have to be modified and lead to other multifractal spectra, as explained in detail
in [33].
The gracious hospitality of the Mittag-Leffler Institute, during the programme Conformal Mappings directed by Peter Jones, and where the work on
the spiral multifractals was initiated in collaboration with Ilia Binder, is gratefully
acknowledged.
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Bertrand Duplantier
Ann. Henri Poincaré
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Bertrand Duplantier
Service de Physique Théorique
Unité de recherche associée au CNRS
CEA, Saclay
F-91191 Gif-sur-Yvette Cedex
France