Exact Multifractal Exponents for Two-Dimensional Percolation

5 Pages·0.17 MB·English

by  Bertrand Duplantier

Exact Multifractal Exponents for Two-Dimensional Percolation
Bertrand Duplantier
Service de Physique Th´eorique de Saclay, F-91191 Gif-sur-Yvette Cedex,
and Institut Henri Poincar´e, 11 rue Pierre et Marie Curie, 75231 Paris Cedex 05, France
(November23, 1998)
9
9
9
fractal structures, such as random walks [8,9], and self-
1 Abstract. Theharmonicmeasure(ordiffusionfield)near
avoiding walks [10]. A further difficulty here is the pres-
n a critical percolation cluster in two dimensions (2D) is con-
ence of a subtle geometrical structure in the percolation
a sidered. Its moments, summed over the accessible external
J cluster hull, recently elucidated by Aizenman et al. [11].
hull, exhibit a multifractal (Mf) spectrum, which I calculate
3 exactly. The generalized dimensions D(n) as well as the Mf Excellent agreement with decade-old numerical data is
function f(α) are derived from generalized conformal invari- obtained, thereby confirming the relevance of conformal
h] ance, and are shown to be identical to those of the harmonic invariancetomultifractality; the exactpredictionforthe
c measure on 2D random or self-avoiding walks. An exact ap- anomalousexponentofapercolativeelectrodegivenhere
e plication to the impedance of a rough percolative electrode also corroborates the multifractal nature of the latter.
m is given. The numerical checks are excellent. Another set of As an illustration of the flexibility of the method, I also
- multifractal exponents is obtained exactly for n independent give the set of exact multifractal exponents correspond-
at self-avoiding walks anchored at therandom fractal boundary ing to the average nth moment of the probability for a
t of a percolation cluster. self-avoiding walk to escape from a percolation cluster
s
. boundary.
t PACS numbers: 05.40.Fb, 05.45.Df, 64.60.Ak, 41.20.Cv
a Consideratwo-dimensionalverylargeincipientcluster
m
, at the percolation threshold p . Define H(w) as the
c
- Percolation theory, whose tenuous fractal structures, Cprobability that a random walker (RW) launched from
d
n called incipient clusters, present fascinating properties, infinity, first hits the outer (accessible) percolation hull
o hasservedasanarchetypalmodelforcriticalphenomena ( ) at point w ( ). We are especially interested in
H C ∈H C
c [1]. The subject has recently enjoyed renewed interest: themomentsofH,averagedoverallrealizationsofRW’s
[
the scaling (continuum) limit has fundamental proper- and
C
1 ties, e.g., conformal invariance, which present a mathe-
v matical challenge [2,3,4]. Almost uncharted territory in = Hn(w) , (1)
08 exact fractal studies is the harmonic measure, i.e., the Zn *wX∈H +
diffusion or electrostatic field near an equipotential ran-
0 where n can be, a priori, a real number. For very large
1 dom fractal boundary, whose self-similarity is reflected
clusters and hulls ( ) of averagesize R, one expects
0 in a multifractal (Mf) behavior of the harmonic measure C H C
these moments to scale as
9 [5].
9
Mf exponents for the harmonicmeasure offractalsare (a/R)τ(n), (2)
/ n
t especially important in two contexts: diffusion-limited Z ≈
a
m aggregation (DLA) and the double layer impedance at where a is a microscopic cut-off, and where the mul-
a surface. In DLA, the harmonic measure actually de- tifractal scaling exponents τ(n) encode generalized di-
-
d termines the growth process and its scaling properties mensions D(n), τ(n) = (n 1)D(n), which vary in a
−
n are intimately related to those of the cluster itself [6]. non-linear way with n [12,13,14,15]. Several a priori re-
o Thedoublelayerimpedanceataroughsurfacebetweena sults are known. D(0) is the Hausdorff dimension of the
c
: goodconductorandanionicmediumpresentsananoma- supportofthemeasure. Byconstruction,H isa normal-
v lous frequency dependence, which has been observed by ized probability measure, so that τ(1) = 0. Makarov’s
i
X electrochemists for decades. It was recently proposed theorem [16], here applied to the Ho¨lder regular curve
r that this is at heart a multifractal phenomenon, directly describing the hull [17], gives the non trivial informa-
a linkedwiththe harmonicmeasureofthe roughelectrode tion dimension τ′(1) = D(1) = 1. The multifractal for-
[7]. Inboththeabovecontexts,percolationclustershave malism [12,13,14,15]further involves characterizing sub-
been studied numerically as generic models. sets α of sites of the hull by a Lipschitz–Ho¨lder
H H
In this Letter, I consider incipient percolation clusters exponent α, such that their local H-measure scales as
α
in two dimensions (2D), and determine analytically the H(w α) (a/R) .The “fractaldimension”f(α) of
∈H ≈
exact multifractal exponents of their harmonic measure. theset α isgivenbythesymmetricLegendretransform
H
I use recent advances in conformal invariance (linked to of τ(n):
quantum gravity), which allow for the mathematical de-
dτ df
scriptionofrandomwalksinteractingwithotherrandom α= (n), τ(n)+f(α)=αn, n= (α). (3)
dn dα
1
Because of the ensemble average (1), values of f(α) n suggests a difference between annealed and apparent
can become negative for some domains of α [18]. quenched averages,as in the DLA case [20].
This Letter is organized as follows: I first present in The first striking observation is that the dimension of
detail the findings and their potential physical signifi- the support of the measure D(0) = D , where D = 7
6 H H 4
cance and applications,before proceeding with the more is the Hausdorff dimension of the standard hull, i.e., the
abstract mathematical derivation. outer boundary of critical percolating clusters [21]. In
Myresultsforthegeneralizedharmonicdimensionsfor fact,D(0)= 4 isthedimensionD oftheaccessible ex-
3 EP
percolation are ternalperimeter[22,11],theotherhullsitesbeinglocated
in deep fjords, which are not probed by the harmonic
D(n)= 12 + √24n+5 1+5, n∈ −214,+∞ , (4) mdeeraivsuedrei.n tItesrmesxaocftrevlaelvuaentDsEcPali=ng 34ophearastobresenderseccreibnitnlyg
(cid:2) (cid:1) path crossing statistics in percolation [11]. In the scal-
valid for all values of moment order n,n > 1 . The
−24 ing continuous regime of percolation, the fjords do close,
Legendre transform (3) of τ(n)=(n 1)D(n) reads
− yielding a smoother (self-avoiding) accessible perimeter
ofdimension 4. This is inagreementwiththe instability
dτ 1 5 1 3
α= (n)= + , (5) phenomenonobservednumericallyonalattice: removing
dn 2 2√24n+1
thefjordswithnarrowneckscausesadiscontinuityofthe
and effectivedimensionofthehullfromDH ≃ 74 toDEP ≃ 34,
whatever microscopic restriction rules are choosen [22].
25 1 α
f(α)= 3 , α 1,+ . (6) In other respects, a 2D polymer at the Θ-point is known
48 − 2α 1 − 24 ∈ 2 ∞ to obey exactly the statistics of a percolation hull [23],
(cid:18) − (cid:19)
(cid:0) (cid:1) andtheMfresults(4-6)thereforeapplyalsotothatcase.
An even more striking fact is the complete identity of
1.5 Eqs. (4-6) to the corresponding results both for random
D(−1/24)=3/2
walks and self-avoiding walks (SAW’s) [10]. In particu-
D(0)=4/3
lar,D(0)= 4 istheHausdorffdimensionofaSAW,com-
3
mon to the external frontier of a percolation hull and of
n) D(1)=1 a Brownian motion [8,9]. Seen from outside, these three
1.0
( D(2)=11/12 fractalcurvesarenotdistinguishedbytheharmonicmea-
D
sure. As we shall see, this fact is linked to the presence
of a universal underlying conformal field theory with a
vanishing central chargec=0.
n* =−1/24
The singularity at α = 1 in the multifractal function
0.5 2
0 1 2 3 4 5 6 7 8 9 10 f(α) is due to points on the fractal boundary where the
n latter has the localgeometry of a needle. Indeed, by ele-
mentary conformal covariance, a local wedge of opening
angle θ yields an electrostatic potential, i.e., harmonic
1.5
measure, which scales as H(R) R−πθ R−α, thus,
formally, θ = π, and θ = 2π cor∼responds∼to the lowest
1.0 f(3)=4/3 possible value αα = 1. The linear asymptote of the f(α)
α) curve for α + ,2f(α) α corresponds to the low-
f( 0.5 est part n →→n∗∞= −214 o∼f t−h2e4spectrum of dimensions.
Its linear shape is quite reminiscent of the case of a 2D
0.0 DLA cluster [24]. Define (H) as the number of sites
N
havingaprobabilityH tobehit. UsingtheMfformalism
0 5 10 15 20 to change from variable H to α (at fixed value of a/R),
α
1/2 showsthat (H)obeys, for H 0,a powerlaw behav-
ior with anNexponent τ∗ = 1+ →lim 1f(α) = 1+n∗.
α→+∞ α
FIG.1. Universal harmonic multifractal dimensions D(n), Thus we predict
andspectrumf(α)ofa2Dincipientpercolationcluster,com-
pared tonumerical results byMeakin et al. (in red). (H) H→0 H−τ∗,τ∗ = 23. (7)
N | ≈ 24
Figure1showstheexactcurveD(n)(4)togetherwith This τ∗ = 0.95833... compares very well with the result
the numerical results for n 2,...,9 by Meakin et τ∗ =0.951 0.030, obtained for 10−5 6H 610−4 [19].
∈ { } ±
al. [19], showing fairly good agreement. The slight up- Letusconsiderforamomentthedifferent,butrelated,
wards move from the theoretical curve at high values of problem of the double layer impedance of a rough elec-
2
trode. In some range of frequencies ω, the impedance latter paths passes only through occupied sites, or only
contains an anomalous “constant phase angle” (CPA) through empty (dual) ones. The probability that the
term (iω)−β, where β <1. It was believed that β would Brownian and percolation paths altogether traverse the
be solely determined by the Hausdorff dimension D(0) annulus (a,R) from the inner boundary circle of ra-
D
of the electrode surface. From a natural RW represen- diusa totheouteroneatdistanceR,i.e.,areina“star”
n
tation of the impedance, a different scaling law was re- configuration ℓ ( ) (Fig. 2), is expected to scale for
S ∧ ∨B
cently proposed: β = D(2) (here in 2D), where D(2) R/a as
D(0) →∞
is the multifractal dimension of the H-measure on the
( n) (a/R)x(Sℓ∧n), (8)
rough electrode [7]. In the case of a 2D porous percola- PR Sℓ∧ ≈
tiveelectrode,ourresults(4)giveD(2) 11,D(0)= 4, n
whence β = 11 = 0.6875. This compare≡s v1e2ry well wit3h where we used Sℓ∧n ≡ Sℓ∧(∨B) as a short hand no-
16 tation, and where x( ℓ n) is a new critical exponent
a numerical RW algorithm result [25], which yields an S ∧
depending on ℓ and n. It is convenient to introduce sim-
emffuelcttiifvraecCtaPlAdeesxcpriopntieonnt[β7]≃. 0.69, nicely vindicating the ilar “surface” probabilities P˜R(Sℓ∧n) ≈ (a/R)x˜(Sℓ∧n)
for the same star configuration of paths, now crossing
Letme nowgivethemainlinesofthederivationofex-
through the half-annulus ˜(a,R) in the half-plane.
ponents D(n) by generalized conformal invariance. We D
focus on site percolationon the 2D triangularlattice; by
universalitytheresultsareexpectedtoapplytoother2D
(e.g., bond) percolation models. The boundary lines of
thepercolationclusters,i.e.,ofconnectedsetsofoccupied
hexagons,form self-avoiding lines on the dual hexagonal R
lattice. Theyobeythestatisticsofloopsinthe (N =1)
O
model,whereN istheloopfugacity,intheso-called“low- R
temperature phase”, a fact we shall recover below [21].
By the very definition of the H-measure, n indepen-
dent RW’s diffusing away from the hull give a geometric
representationofthenth momentHn,forninteger. The
values so derived for n N will be enough, by convexity
∈
arguments, to obtain the analytic continuation for arbi-
traryn’s. Figure2 depictsn independent randomwalks,
inabunch,firsthittingthe externalhullofapercolation
cluster at a site w = ( ). As explained in ref. [11], such
•
a site, to belong to the accessible hull, must remain, in
the continuous scaling limit, the source of at least three
R
non-intersecting crossing paths, noted S3, reaching to a FIG. 2. An “active” site (•) on the accessible external
(large) distance R. These paths are “monochromatic”:
perimeter for site percolation on the triangular lattice. It
one path runs only through occupied (light blue) sites; is defined by the existence, in the scaling limit, of ℓ = 3
theothertwo,dual lines,runthroughempty(white)sites non-intersecting, and “monochromatic” crossing paths S
3
[11]. Thedefinitionofthestandardhullrequiresonlythe (dotted lines), one on the incipient (light blue) cluster, the
origination, in the scaling limit, of a “bichromatic” pair other two on the dual empty (white) sites. The points ⊙ are
of lines [11]. Points lacking additional dual lines are entrancesoffjords, whichcloseinthescalinglimit andwon’t
2
not acceSssible to RW’s after the scaling limit is taken, support the harmonic measure. Point (•) is first reached by
three independent RW’s (red, green, blue), contributing to
because their (white) exit path becomes a straitpinched
H3(•). The hull of the incipient cluster (golden line) avoids
by other parts of the (light blue) occupied cluster. The
the outer frontier of the RW’s (thick blue line). A (random)
bunch of independent RW’s avoids the occupied cluster, R
Riemann map of the latter onto the real line reveals the
and defines its own envelope as a set of two boundary
presenceofanunderlyingℓ=3path-crossingboundary oper-
lines separating it from the occupied part of the lattice, ator, i.e, of an L=4-line, or two-cluster, boundary operator
thus from 3 (Fig. 2). in the associated O(N = 1) model, with dimension in the
S
Let us introduce the notation A∧B for two sets, A, half-plane x˜ℓ=3 =x˜OL=(Nℓ+=11=)4 =x˜Ck=2 =2. Both accessible hull
B, of random paths, conditioned to be mutually avoid- and Brownian pathshavea frontier dimension 4.
3
ing, and A B for two independent, thus possibly inter-
∨
secting, sets [10]. Now consider n independent RW’s, or When n 0, ( ) resp. ˜ ( ) is the prob-
R ℓ R ℓ
Brownian paths in the scaling limit, in a bunch noted → P S P S
( )n, avoidingBa set ( )ℓ of ℓ non-intersecting ability of having ℓ simul(cid:16)taneous mono(cid:17)chromatic(non-
ℓ
∨B S ≡ ∧P intersecting) path-crossings traversing the annulus in
crossing paths in the percolation system. Each of the
the plane (resp. half-plane), with associated exponents
3
x x( 0) and x˜ x˜( 0) [3,11]. These expo- random paths near their random frontier, i.e., the prod-
ℓ ℓ ℓ ℓ
≡ S ∧ ≡ S ∧
nentshavebeenstudiedinref.[11],andshownrigorously uct of two “boundaryoperators”onthe randomsurface.
to be actually independent of the coloring of the paths, The latter sum is mapped by the functions U, V, into
withthe restrictioninthebulk thatthereexistatleasta the scaling dimensions in R2 [10]. The structure thus
path on occupied sites and one on dual ones, thus ℓ>1. unveiled is so stringent that it immediately yields the
Here the exponents appear as analytic continuations of values of the percolationcrossingexponents x ,x˜ ofref.
ℓ ℓ
the harmonic measure ones (8) to n 0, and should [11],andourharmonicmeasureexponentsx( n)(8).
ℓ
→ S ∧
correspond to the definitions of ref. [11]. First, for a set = ( )ℓ of ℓ crossing paths, we have
ℓ
S ∧P
In terms of definition (8), the harmonic measure mo- from the recurrent use of (11)
ments (1) simply scale as R2 ( n) [18],
n R ℓ=3
which, combined with Eqs.Z(2)≈and (8P), leSads t∧o xℓ =2V ℓU−1(x˜1) , x˜ℓ =U ℓU−1(x˜1) . (13)
τ(n)=x( n) 2. (9) For percolati(cid:2)on, two va(cid:3)lues of half-(cid:2)plane cross(cid:3)ing expo-
3
S ∧ −
nents x˜ are known by elementary means: x˜ = 1,x˜ =
ℓ 2 3
Using the fundamentalmapping of the conformalfield 2[3,11].From(13)wethusfindU−1(x˜ )= 1U−1(x˜ )=
theory (CFT) in the plane R2, describing a critical sta- 1U−1(x˜ )= 1, (thus x˜ = 1 [27]), whi1ch in2turn giv2es
3 3 2 1 3
tistical geometrical system, to the CFT on a fluctuat-
ing abstract random Riemann surface, i.e., in presence 1 ℓ
x =2V 1ℓ = ℓ2 1 ,x˜ =U 1ℓ = (ℓ+1).
ofquantum gravity [26],Ihaverecentlyshownthat there ℓ 2 12 − ℓ 2 6
existtwouniversalfunctionsU,andV,dependingonlyon (cid:0) (cid:1) (cid:0) (cid:1) (cid:0) (cid:1)
Wethusrecovertheidentity,previouslyrigorouslyestab-
thecentralchargecoftheCFT,whichsufficetogenerate lishedinref.[11],ofx =xO(N=1),x˜ =x˜O(N=1)withthe
all geometrical exponents involving mutual avoidance of ℓ L=ℓ ℓ L=ℓ+1
L-lineexponentsoftheassociated (N =1)loopmodel,
random star-shaped sets of paths of the critical system O
inthe “low-temperaturephase”. ForLeven,these expo-
[10]. For c=0, which corresponds to RW’s, SAW’s, and
nents also governthe existence of k= 1L spanning clus-
percolation, these universal functions are: 2
ters [21,11], with the identity xC =x = 1 4k2 1
U(x) = 1x(1+2x), V (x)= 1 4x2 1 . (10) in the bulk [21], and x˜Ck = x˜ℓ=2kk−1 =ℓ=132kk(2k12−(cid:0)1) in−th(cid:1)e
3 24 − half-plane [21,28,29]. The non-intersection exponents of
(cid:0) (cid:1) k′ Brownian paths arealsogivenbyxℓ,x˜ℓ forℓ=2k′ [8],
withV (x) U 1 x 1 . Considernowtwoarbitrary
≡ 2 − 2 so we observe a complete equivalence between a Brown-
random sets A,B, involving each a collection of paths ian path and two percolating crossing paths, in both the
(cid:0) (cid:0) (cid:1)(cid:1)
in a star configuration, with proper scaling crossing ex-
plane and half-plane.
ponents x(A),x(B), or, in the half-plane, crossing ex-
For the harmonic exponents in (8), we fuse the two
ponents x˜(A),x˜(B). If one fuses the star centers and n
objects and ( ) into a new star n (see Fig.
ℓ ℓ
requires A and B to stay mutually avoiding, then the S ∨B S ∧
2), and use (11). We just have seen that the boundary
new crossing exponents, x(A B) and x˜(A B), obey ℓ-crossing exponent of , x˜ , obeys U−1(x˜ ) = 1ℓ. The
the star algebra [8,10] ∧ ∧ Sℓ ℓ ℓ 2
bunch of n independent Brownian paths have their own
n
x(A B) =2V U−1(x˜(A))+U−1(x˜(B)) half-plane crossing exponent x˜((∨B) ) = nx˜(B) = n,
∧ since the boundary dimension of a single Brownianpath
x˜(A B) =U U(cid:2)−1(x˜(A))+U−1(x˜(B)) (cid:3), (11) is trivially x˜( )=1 [8]. Thus we obtain
∧ B
where U−1(x) is th(cid:2)e inverse function of U (cid:3) x( n)=2V 1ℓ+U−1(n) . (14)
Sℓ∧ 2
U−1(x)= 1 √24x+1 1 . (12) Specifying to the case ℓ=(cid:0)3 finally gives(cid:1)from (10)(12)
4 −
(cid:0) (cid:1) 1 5
If, on the contrary, A and B are independent and x( 3 n)=2+ (n 1)+ √24n+1 5 ,
S ∧ 2 − 24 −
canoverlap,thenby trivialfactorizationofprobabilities,
(cid:0) (cid:1)
x(A B)=x(A)+x(B),andx˜(A B)=x˜(A)+x˜(B) from which τ(n) (9), and D(n) Eq.(4) follow, QED.
∨ ∨
[10]. Therules(11),whichmixbulkandboundaryexpo- This formalism immediately allows many generaliza-
nents, can be understood as simple factorizationproper- tions. For instance, in place of n random walks, one can
tiesonarandomRiemannsurface,i.e.,inquantumgrav- consider a set of n independent self-avoiding walks ,
ity[8,10],orasrecurrencerelationsinR2 betweenconfor- whichavoidtheclusterfractalboundary,exceptforthePir
mal Riemann maps of the successive mutually avoiding commonanchoringpoint. Theassociatedmultifractalex-
paths onto the line R[9]. On a random surface, U−1(x˜) ponents x( (( )n) are given by the same formula
ℓ
is the boundary dimension corresponding to the value x˜ (14), with Sthe∧arg∨uPment n in U−1 simply replaced by
in R R+, and the sum of U−1 functions in Eq. (11) the boundary scaling dimension of the bunch of inde-
repre×sentslinearly the juxtapositionA B oftwo sets of pendent SAW’s, namely [10] x˜(( )n) = nx˜( ) = n5,
∧ ∨P P 8
4
for ℓ = 3. These exponents govern the universal multi- (1991);
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Acknowledgements: I thank M. Aizenman, D.
Kosower,and T. C. Halsey for fruitful discussions.
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5