stochastic geometry in nature: fractals and multifractals...

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Stochastic geometry in nature: Stochastic geometry in nature: fractals and multifractals and how to study themfractals and multifractals and how to study them

Il A G bIlya A. GruzbergUniversity of Chicago

Colloquium, Texas A&M University , April 17, 2008

Random shapes in nature: growth patternsRandom shapes in nature: growth patterns

• Hele-Shaw flow

• Real patterns in nature:

Hele Shaw flow• Mineral dendrites• ElectrodepositionElectrodeposition


Random shapes in nature: thin filmsRandom shapes in nature: thin films

• AFM image of WO thin film3


Random shapes in nature: thin filmsRandom shapes in nature: thin films

• Height clusters and isolines

• Positive heights in color,negative heights black

• Boundaries of positive height clusters


negative heights black height clusters

Random shapes in nature: turbulenceRandom shapes in nature: turbulence

• Inverse cascade in 2D Navier-Stokes turbulence

0.6

0.7

0.8

0.3

0.4

0.5

0

0.1

0.2

-0.8 -0.6 -0.4 -0.2 0 0.2

• Vorticity clusters

-0.8 -0.6 -0.4 -0.2 0 0.2

• Zero vorticity lines


Vorticity clusters Zero vorticity lines

Random shapes in nature: turbulenceRandom shapes in nature: turbulence• Surface quasigeostrophic (atmospheric) turbulence

(a)

x0 x∞

• Temperature clusters • Temperature isolines

Th h bl l ti l t


• These shapes resemble percolation clusters

Random shapes in 2DRandom shapes in 2D

• Two classes:

• Critical clusters in equilibrium statistical mechanics

Local Boltzmann weights or probabilities

• Non-equilibrium growth patterns

• Similar complexity but very different understanding

Non-local, history-dependent rules

• Similar complexity but very different understanding


Critical clustersCritical clusters

• Percolation clusters (site percolation)

Every site is either ON (black) with probabilityEvery site is either ON (black) with probabilityor OFF (white) with probability



• Larger critical percolation clusters



• Ising spin clusters


Other equilibrium clustersOther equilibrium clusters

• q-states Potts model: q = 2 – Ising, q = 1 – percolation

• Can continue


Other equilibrium clustersOther equilibrium clusters

• O(n) loop models: ( ) pn = 0 – self-avoiding walks (polymers)

• Can continue

• Solid-on-solid (SOS) models, roughening transitions,( ) , g g ,dimers, random tilings,…


DLA appletDLA applet

• Show DLA applet from S ow app et o

http://apricot.polyu.edu.hk/~lam/dla/dla.html


DLA appletDLA applet


Growth patternsGrowth patterns

• Diffusion-limited aggregation T. Witten, L. Sander, 1981

• Many different variants


• Many different variants

Viscous fingering vs. DLAViscous fingering vs. DLA

O. Praud and H. L. Swinney, 2005


Random shapes: stochastic geometryRandom shapes: stochastic geometry

• Fractal properties

• Multifractal spectrum of harmonic measure

• Crossing probability

• Left vs. right passage probabilityLeft vs. right passage probability

• Many more …


Stochastic geometry: fractal propertiesStochastic geometry: fractal properties

• Fractal properties B. Mandelbrot, 1967

• Dimensions (critical exponents)

• Correlation functions


Stochastic geometry: fractal propertiesStochastic geometry: fractal properties

• Subsets and boundaries with different fractal dimensions

Backbone, danglingends, minimal path

Hull and externalperimeter


e ds, a pat pe ete

Stochastic geometry: multifractal exponentsStochastic geometry: multifractal exponents

• Lumpy charge distribution on a cluster boundary

• Cover the curve by small discsof radius

• Charges (probabilities) inside discs

• Moments

• Non-linear is the hallmark of a multifractal


Harmonic measure on DLA clusterHarmonic measure on DLA cluster


Stochastic geometry: multifractal exponentsStochastic geometry: multifractal exponents• Multifractal measures: electric field of a charged cluster


2D critical phenomena2D critical phenomena• Scale invariance L.Kadanoff, A. Patashinski, V. Pokrovsky, 1966

Critical fluctuations (clusters) are self-similar at all scalesThe basis for renormalization group approach

• Conformal invariance A. Polyakov, 1970

• In 2D conformal maps = analytic functions

0.4

0.5

3

−0.1

0

0.1

0.2

0.3

−1

0

1

2

−0.6 −0.4 −0.2 0 0.2 0.4 0.6

−0.5

−0.4

−0.3

−0.2

−3 −2 −1 0 1 2 3 4 5

−3

−2

−1


2D critical stat mech models2D critical stat mech models

• Traditional approach: A. Belavin, A. Polyakov, A. Zamolodchikov, 1984

Conformal field theory: correlations of local observablesImportant parameter: central charge

• New focus:

Stochastic geometry: non local structures clusterStochastic geometry: non-local structures – clusterboundaries, their fractal and global properties

Finite geometries: conformal invariance made precise


Crossing probabilityCrossing probability• Crossing probabilities J. Cardy, 1992

Is there a left to right crossing of white hexagons?


Crossing probabilityCrossing probability

And now?


Cluster boundariesCluster boundaries

• Focus on one domain wall using certain boundary conditions

• Conformal invariance allows to consider systems in simple• Conformal invariance allows to consider systems in simple domains, e. g. upper half plane


Exploration processExploration process

• Cluster boundary can be “grown” step-by-step

h i d i d b l l i• Each step is determined by local environment

?

• Description in terms of differential equations in continuum


Exploration processExploration process

• Cluster boundary as a stochastic growth process: percolation


Conformal mapsConformal maps

• Specify a 2D shape by a function that maps it to a simple shape

• Always possible by Riemann’s theorem


Loewner equationLoewner equationC Loewner 1923

• Describes upper half plane with a cut along a curve

C. Loewner, 1923

pp p g


Loewner equationLoewner equation

• Example:

• In general


Loewner equation: sketch of derivationLoewner equation: sketch of derivation


Loewner equationLoewner equation

• Properties of the trace are quite amazing:

• If is smooth, never intersects itself

• If is singular enough, may touch itself or the boundary at double points

,

or the boundary at double points


SLE postulatesSLE postulates

• Conformal invariance of the measure on curves


Lattice models: domain Markov propertyLattice models: domain Markov property

b bb

c

b

c

a a


SLE postulatesSLE postulates

• Markov property and stationary increments

i d d f ( k ) d• independent from (Markov property) andhas the same distribution as


SchrammSchramm--Loewner evolutionLoewner evolution

• Conformal invariance leads uniquely to Loewner equation

O. Schramm, 1999

driven by a Brownian motion:

• Noise strength is an important parameter:

SLE generates conformally-invariant random curvesstatistically equivalent to critical cluster boundaries


Properties of SLEProperties of SLES R hd d O S h 2001

• is a simple curve for

S. Rohde and O. Schramm, 2001V. Beffara, 2002

• touches boundary and itself for

• densely fills a 2D region for

γ γ(t ) (t ) KKt

t

densely fills a 2D region for

κκ κ< 40 < 4 < < 8 > 8

Kt

κκ κ< 40 < 4 < < 8 > 8

• Fractal dimension of is


SLE versus CFTSLE versus CFT

• Conjecture: SLE describes all critical 2D systems withM. Bauer and D. Bernard, 2002

R. Friedrich and W. Werner, 2002

• Relation between noise strength and central charge:


SLE versus CFTSLE versus CFT

• Duality

B. Duplantier, 2000

Duality

ForK

describes its perimeter

describes the traceγ(t ) Kt

describes its perimeter

• Example: percolation hull with

κ4 < < 8

Example: percolation hull withwhile external perimeter with


SLE versus CFT: examplesSLE versus CFT: examples

• Loop-erased random walk• Self-avoiding walk

• Crystal facets, dimers,level lines of Gaussian field

• Ising model spin clusters

level lines of Gaussian field

• Percolation hull• Ising model FK clusters

• O(n) model and q states Potts models are related to SLE via

Percolation hull• Uniform spanning trees

• O(n) model and q-states Potts models are related to SLE via


A path in a uniform spanning tree: loop-erasedrandom walkrandom walk


Self-avoiding walk


Double domino tilings


Level lines of Gaussian free field


Boundary of a percolation cluster


Random Peano curve


Calculations with SLECalculations with SLE

• SLE as a Langevin equation

• Shift

• Simple way of deriving crossing probabilities various

• Langevin dynamics diffusion equation

• Simple way of deriving crossing probabilities, various critical exponents and scaling functions

• Multifractal spectra for critical clusters


p

Crossing probabilityCrossing probabilityJ. Cardy, 1992


Crossing probabilityCrossing probability

C

L. Carleson

S. Smirnov, 2001

x

X

r “Most difficult theorem

A B1

r Most difficult theoremabout the identity function”

P. Jones


LeftLeft--passage probabilitypassage probability

• Take in the upper half plane

O. Schramm, 2001

• SLE trace passes to the left of with probability


Conformal multifractalityConformal multifractality

B. Duplantier, 2000• Originally obtained by quantum gravity

• For critical clusters with central charge

• Generalization to include winding of curves I Binder 1998

• Can obtain from SLE, somewhat difficult…

Generalization to include winding of curves I. Binder, 1998B. Duplantier, I.Binder, 2002

• Can now obtain this and more using traditional CFTE. Bettelheim, I. Rushkin, IAG, and P. Wiegmann, 2005

A Belikov IAG I Rushkin 2008


A.Belikov,IAG, I.Rushkin, 2008

Variants and generalizations of SLEVariants and generalizations of SLE

• Different geometries: radial and others

• Radial SLE gives curves growing from the unit circleto the origin (or infinity):



• Multiple SLEJ. Cardy, 2003

M. Bauer, D. Bernard, K. Kytola, 2005

• Dyson Brownian motion (random matrices)• Dyson Brownian motion (random matrices)

• Calogero-Sutherland models



SLE f CFT i h i i d

E. Bettelheim, IAG, A.W.W. Ludwig and P. Wiegmann, 2005

• SLE for CFT with continuous symmetries and (Wess-Zumino models)

• SLE trace acquires “twisting” (a spin) described by an additional noise



• SLE driven by Lévy processes: branching growth

I. Rushkin, P. Oikonomou, L. P. Kadanoff and IAG, 2005



• SLE driven by Lévy processes: global properties

P. Oikonomou, I. Rushkin, IAG and L. P. Kadanoff, 2007

X (t ) » t 1=®; Y (t ) » A + B t 1¡ 1=®


Applications of SLEApplications of SLE• 2D turbulence D. Bernard, G. Boffetta, A. Celani, G. Falkovich, 2006

• Zero vorticity contours are with · ¼ 5:9(percolation?)

0.06 (κ t)1/2 P(ξ(t))

0.03

0.04

0.05

0.06

< ξ(t)2 >

( t) P(ξ(t))

0.01

0.02

0.03

456789

<ξ(t)2>/t-3 -2 -1 0 321

ξ(t)/(κ t)1/2

00 0.005 0.01

t

4

0 0.005 0.01


Applications of SLEApplications of SLE• 2D turbulence D. Bernard, G. Boffetta, A. Celani, G. Falkovich, 2006

•Temperature isolines are with · = 4 § 0:2(Gaussian free field?)


Applications of SLEApplications of SLE

• 2D quantum chaosJ. P. Keating, J. Marklof, and I. G. Williams, 2006

E. Bogomolny, R. Dubertrand, C. Schmit, 2006

N d l li f h i f i i h• Nodal lines of chaotic wave functions are with (expect percolation ) · = 6· ¼ 5:3 (6:05; 5:92)


Applications of SLEApplications of SLE• 2D Ising spin glass

C. Amoruso, A. K. Hartmann, M. B. Hastings, M. A. Moore, 2006D Bernard P Le Doussal A Middleton 2006D. Bernard, P. Le Doussal, A. Middleton, 2006

• Domain walls are with(somewhat disappointing: nothing special about this value)

· ¼ 2:1 (2:32 § 0:08)(somewhat disappointing: nothing special about this value)

4

0.05

L=W=400, R=50, F-APL=W=720, R=90, F-APL=W=720, R=360, F-APL=128, W=1024, R=64,

2

3

(2t)

-ξ2 (t

)]/t

L=256, from constrained endL=128, from constrained end200

3000

0.05

P(φ

)-P

2(φ)

L=128, W=1024, R=64,L-AP (fixed end)Pκ=2.24

(φ)-Pκ=2(φ)

Pκ=2.32(φ)-Pκ=2

(φ)

Pκ=2.40(φ)-Pκ=2

(φ)

Pκ=2.85(φ)-Pκ=2

(φ)

φ

100

101

102

103

104

105

0

1

[ξ2 (

L=128, from constrained endL=128, floating BCsL=64, floating BCs

0 100 200x

0

100

200y

0 0.25 0.5 0.75 1

-0.05

P

R

Y

X


10 10 10 10 10 105

t0 0.25 0.5 0.75 1

φ/π


mX d

• Length of SLE: fractal variation T. Kennedy, 2006

` ´i = 1

jz( t i + dt ) ¡ z( t i ) jdf


Applications of SLEApplications of SLE• First passage times along fractal curves

A. Zoia, Y. Kantor, M. Kardar, 20071

0.5y

1 1

−1 −0.5 0 0.5 1 0

x

y

1

y

1

−0.5 0.50

−0.5 0.50


x−0.5 0.5

x−0.5 0.5


• Morphology of thin ballistically deposited films

0.3

⟨δξ(

0)δξ

(t)⟩

0.0x10+00

1.0x10-06

0.2

0.3

P(ξ

(t))

t0 0.05 0.1

-1.0x10-06

⟨ξ(t)

2 ⟩

0.1

0.2

ξ(t)/(κt)1/2

(κt)

1/2

P

-1 0 1

0 0.02 0.04 0.06 0.08 0.10

0.1

t0 0.02 0.04 0.06 0.08 0.1

• Height isolines are with (Ising)· = 3 § 0:2


g ( g)§


• Morphology of KPZ surfaces

• Height isolines are with (self-avoiding random walk)

· = 8=3 § 0:1


(self-avoiding random walk)

What about growth patterns?What about growth patterns?

• Both for DLA and Hele-Shaw patterns

M ltifractal spectr m is kno n n mericall• Multifractal spectrum is known numerically

• Very few analytical results


Continuous Loewner chainsContinuous Loewner chains

• Growth along the whole boundary of a domain


ExamplesExamples

• Radial (multiple) SLE for

• Laplacian growth for Integrable model with finite time singularitiesg g

• Dielectric breakdown for

• Hele-Shaw flow with surface tension

• Other regularized Laplacian growth problems


Integrability of Laplacian growthIntegrability of Laplacian growth

I. Krichever, M. Mineev-Weinstein, P. Wiegmann, A. Zabrodin

S. Richardson, B. Shraiman and D. Bensimon

• LG conserves exterior harmonic moments of the interface

• Families of explicit solutions• Families of explicit solutions

• Integrable model related to random matrices

• Loewner equation appear as a reduction


Discrete Loewner chainsDiscrete Loewner chains

• Iterated conformal maps M. Hastings and L. Levitov, 1998


Discrete Loewner chainsDiscrete Loewner chains

M. Stepanov and L. Levitov, 2001

• Excellent tool for generating and studying DLA-like patternsExcellent tool for generating and studying DLA like patterns

• Variants and generalizations: interpolations between DLA and LG Laplacian random walks


and LG, Laplacian random walks, …

ChallengesChallenges

• Applications of SLE to disordered systems: spin glasses,electronic localization and quantum Hall transitions

• Application to non-equilibrium systems: turbulence,force chains in jammed granular matterj g

• Analysis of Loewner chains and applications to growth phenomena: Laplacian growth DLA fractures etcphenomena: Laplacian growth, DLA, fractures, etc.

• Combining integrability with stochastic dynamics?

• 3D random shapes: conformal maps are not useful…


3D DLA3D DLA


• From http://math.mit.edu/~chr/research/3d-dla.htm

ThanksThanks

M. BauerR. Bauer

G. LawlerA. Ludwig

A. BelikovD. BeliaevD Bernard

gN. MakarovB. Nienhuis P OikonomouD. Bernard

E. BettelheimI. Binder

P. OikonomouI. RushkinO. Schramm

J. CardyB. DuplaniterL. Kadanoff

S. SheffieldS. SmirnovP. Wiegmann

W. KagerT. Kennedy

gT. Witten


stochastic geometry in nature: fractals and multifractals...

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