Stochastic Loewner Evolution - AUusers-phys.au.dk/fogedby/statphysII/notes/SLE2.pdf · Padova April...

Stochastic LoewnerEvolution

Fractal shapes and curves in 2D

Hans FogedbyAarhus University

andNiels Bohr Institute

Padova April 27, 2007 Stochastic Loewner Evolution 2

Some players on the SLE sceneOded Schramm

Wendelin WernerFields Medal 2006

Gregory LawlerBernhard Nienhuis

Leo KadanoffJohn Cardy

Ilya Gruzberg

Stanislav Smirnov

http://research.microsoft.com/~schramm/IMGP0845.JPG

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OutlineFractal shapes in 2DLattice models Fractal dimensionScaling behaviorScaling limitConformal transformationsRandom curvesLoewner evolution (LE)Stochastic Loewner evolution (SLE)Phases of SLE and fractal dimensionsSummary and conclusion


Fractal shapes in 2DRandom walk Percolation cluster at

critical concentration

Diffusion limitedaggregation

Ising cluster atcritical temperature

http://images.google.dk/imgres?imgurl=http://www.thp.uni-koeln.de/~dt/science/percolation/leath75.png&imgrefurl=http://www.thp.uni-koeln.de/~dt/science/percolation/index.html&h=454&w=536&sz=14&hl=da&start=2&tbnid=iGbVgL26K_DIrM:&tbnh=112&tbnw=132&prev=/images%3Fq%3Dpercolation%2Bcluster%26svnum%3D10%26hl%3Dda%26lr%3D%26sa%3DG


Lattice modelsRandom walk

Percolation

Ising model

Random walk on square lattice


Percolation

Occupy site or bond with probability p, 0<p<1At critical concentration p=pc , infinite cluster

Percolation is the phenomenon oftransport of fluid through a porous medium


Clusters at critical concentrationCritical spanning cluster (blue) Self similarity


Percolation threshold


Ising modeli Local degree of freedom , lattice point

Hamiltonian:

coupling constant nearest neighbor coupling Probability

1

of

c

( { })

onfiguration < >

( {

:

} )

i j

i

H Jij

Jij

P

σ

σ σ σ

σ

= ±

= −< >

=

∑

Partition function: , temperature

Magnetization (order parameter

e

)

xp( ({ }) / )

exp( / )

:

{ }

1 exp( / ){

}

i i

H kTZ

Z H kT

i

m H TZ

i

T

σ

σ

σ σσ

−

= −

=< >= −

∑

∑


Fractal dimension Mandelbrot and Nature"Clouds are not spheres, mountains are not cones,coastlines are not circles, and bark is not smooth,nor does lightning travel in a straight line.“

Mandelbrot, 1983.

New geometrical description of scale invariant objectsin natural sciences and mathematicsFractals characterized by non integer dimension


Fractal dimensionCover object with N boxes of size aN(a) will depend on a as a power

- ln ln

DN aNDa

=

= −

D is the fractal dimension


Fractal dimensionThe Koch curveN(a)=a-D, D fractal dimension

N(a=1) = 1, N(a=1/3) = 4, N(a=1/9)=16, ..N(a=1/3n)=4n

DKoch =log4/log3 ~ 1.261< DKoch <2

Koch kurve is self-similar


Scaling behavior

Random walkIsing modelPercolation


Random walk is scale invariant<- R ->

R is size of clusterN is number of stepsone step pr time unitN proportional to time T

Scaling:

scaling exponent =1/2R N T Tυ

υ∝ ∝ =

Self similarity


Ising model is scale invariant at Tc

Monte Carlo simulation Ising-Model

2nd order phase transition at TcVanishing order parameter mLong range spin orderDiverging correlation lengthInfinite Ising cluster

http://bartok.ucsc.edu/peter/java/ising/isingAppletLarge.html


Self similarity at TcT=0.99 Tc T=Tc T=1.22 Tc


Critical properties at Tc

Magnetization m is the order parameterm~|T-Tc|β, β critical exponent

Susceptibility χ characterizes response of m to hχ=dm/dh~|T-Tc|-γ, γ critical exponent

Correlation length ξ characterises size of domainsξ~ |T-Tc|-ν, ν critical exponent


Percolation is scale invariant at pc

P∞ is probability of belonging to infinite clusterP∞ is the order parameterP∞ ~|p-pc|β, β critical exponent

Correlation length ξ characterisessize of clustersξ~ |p-pc|-ν, ν critical exponent

S is mean cluster sizeS~|p-pc|-γ, γ critical exponent


Scaling limit (continuum limit)Lattice shrinks (continuum limit)Site variables becomes fieldsLattice models become field theoriesLattice models at criticality becomeconformal field theoriesIssue: How to understand scaling in thecontinuum limit


Brownian motionContinuum limit of random walk is Brownian motion


Schematic Brownian motion

2D computer simulationBlue particle pollen grainRed particle representmolecules colliding withpollenPollen grain performsBrownian motion


Scaling limit of random walkRandom walk Brownian motionS(n+1):=S(n)±1, S(0)=0

n number of steps (time t)δ is scale parameter δ →0

Random walk S(n) converges to Brownian motion B(t) (BM)

BM is a continuous curveBM is not differentiableBM is scale invariantBM is at a critical pointBM has fractal dimension D=2BM is plane-filling

BM is a Markov processStatistical properties:

<B(t)>=0, <B(t)2>= κt <(B(t)-B(s))2>=κ |t-s|


Scaling limit of Ising modelIn scaling limit the Ising model becomes a Φ4 - field theoryAt the critical point renormalization group (RG) methods yield exponentsThe local field and itscorrelations are the central objectsRG method does not describecritical domains etcRG method does not give theGEOMETRY at criticality

i

2 2 4 2

Site variable local field ( )Hamiltonian Landau functional

- ( )

Partition function path integral

exp( / ) exp( ){ }

i jij

r

r

H J F R U d r

Z H kT D F

i

σ

σ σ

σ

< >

→→ Φ

→

⎡ ⎤= → = ∇Φ + Φ + Φ⎣ ⎦

→

= − → Φ −

∑ ∫

∑ ∏∫

Renormalization groupKen Wilson

Nobel prize 1982


The issueInvestigate critical behavior directly in thecontinuum limitFocus on critical domains and curvesAssume conformal invariance at criticalityConformal invariance for field theory yieldsconformal field theory (massless)


Conformal transformations

Rotation

Dilatation

Shear


Conformal transformations

By composing maps:Any shape can be mappedto any other shapeIn 2D geometry is thesame as complex analysis

Conformal transformation:Local translationsRotationsDilatationsANGLES PRESERVED

Riemann’s theorem: A shape without holes in plane zcan be mapped to the unit diskin plane w by means of an analyticfunction:

w= g(z)

z plane w plane


Examples (transformation of grid)

Conformal transformation


Random curves as exploration processPercolation: Completed process: Critical percolation interface:

Percolation growth process: Large percolation growth process in UHP:


Random curves as exploration processIsing model on honeycomb lattice:

Exploration process for Ising model:

Loop erased random walk:


Loewner evolution (LE)Loewner evolution addresses issue of generatingcurves in 2D in the continuum limitIdea: Define gradual conformal transformation g(z)Change shapes by changing conformaltransformationParametrize transformation by ”time” tAssume identity transformation at infinityMap shape to half plane (chordal LE)


Mapping shape by Riemann’s theoremHalf plane HChanging shape Kt (the hull)Complement to shape H\KtComplement H\Kt mapped to H by analytic function gtReal axis plus boundary of Kt mapped to real axis


Boundary conditions

0( )tg z z= =

2( ) (1/ )tt

Cg z z O zz

= + +

The identity map at infinity in H

Ct is the capacityParametrization

The identity map at the initial time t=0

2tC t=


The growing stick2

2

( ) 4

( ) 4 , inverse mapt

t

w g z z t

z f w w t

= = +

= = −

C at (0,2 ) in plane

B and D at ( 2 ,0) in plane

i t z

t w∓

gt(z)


Composed maps

By composing mapswe can generatevarious shapes


Equation of motion for growing stick

2

Growing stick map:

( ) 4 tw g z z t= = +

Equation of motion:( ) 2

( )t

t

dg zdt g z

=


The Loewner equation

1

Inverse map ( ) analytic in HDispersion relation (Cauchy's theorem)

( ')( ) ( ) ', 0'

is spectral weight Half plane capacity

( ') '

For

( ) , ( )

Loewner ti

t

tt t t

t

t t

t tt t

f w

xf w g w w dxw x

C x dx

zC Cf w w g z zw z

ρ ρ

ρ

ρ

−= = − ≥−

=

→∞

− +

∫

∫

∼ ∼

me/ 2tt C=

2 1 1

1

2 1 2

Capacity additive for composed maps2 2( )( ( )) ( ) .. ..

( )t t tt

t t tg g z g z zg z z

++ + + +∼ ∼


The Loewner equation'

1

Assume shape shape for '' , ( \ )

Sequence of maps and 'swallows' shape

'swallows' small bump

( ) ( ( )) or ( ) ( ( ))

Dispersion law for

t t

t t t t t

t t

t t

t t

t t t t t t t t

K K t tt t t K g K K

g gg K

g K

g z g g z g z g g z

δ

δ δ

δ δδ

δ δ

δ δ

+

−+ +

⊂ <= + =

= =1

1

:( ') '( )

'( ') '( ) ( )( ) '

set and let 0( ) ( ') '

( ) 'measure ( ) determines how andwhere the shape grows

t

tt

tt t t

t t

t t

t t

t

t

t

gx dxg w w

w xx dxg z g z

g z xt t

dg z x dxdt g z x

x dxK

δδ

δδρδ

δρ

δρ υ δ δυ

υ

−

−

++

= −−

= −−

→

=−

∫

∫

∫

∼0

Growth at a point ( ) 2 ( ) image of growing tip under ( )

Standard Loewner equation2 ( ) , ( )

( )

t t

t t

tt t

x xg z

d g z g z zdt g z

υ δ ξξ

ξ

= −

= =−


The Loewner equation

Riemann’s theorem


Loewner evolution, examples

Constant forcing

Linear forcing

Periodic forcing Growing periodic


Loewner evolution - summary

Time evolution of map gt(z) from z to wCurve γ(t) in 2D driven by real function ξ(t)Tip of curve zc(t) given by ξ(t)= gt(zc(t))ξ smooth - curve γ non-intersectingξ periodic - curve γ self-similarξ singular - curve γ self-intersecting at finite time


Stochastic Loewner evolution (SLE)Schramm Loewner EvolutionIdea: Grow domain wall step by step (Markov process)

Implement Markov property in the continuum limit

μ is the probability measure on γ


Stochastic Loewner evolution (SLE)Implement conformalinvariance to ensure scaling

Transformation of probability measure(invariant under conformal tranport)

Conformal transformation to upperhalf plane, reference plane (chordal SLE)


Stochastic Loewner evolution (SLE)

Loewner equation (1923):

Schramm (1999):

Bt: Brownian motion, <(Bt - Bs)2>= κ |t-s|Parameter κ, SLEκ

Non self-crossing curves on the half plane from a0 to ∞


SLE, summary

Drive curve γ(t) by random function ξ(t)Curve is fractal if ξ(t) is a Brownian motion< (ξ(t)- ξ(s))2 >= κ |t-s|κ is SLE parameter, SLEκ0< κ <4: curve non-intersecting (1<D<3/2)4< κ <8: curve intersecting (3/2<D<2)κ >8: curve space filling (D=2) Fractal dimension:

1 8

D κ= +


The phases of SLEκ

Computer simulation of SLEκ (Vincent Beffara)


Phases of SLEκ

Non-intersecting fractal curveFractal dimension D=5/4

Self-intersecting fractal curveFractal dimension D=7/4


Phases of SLEκ

Space filling fractal curveFractal dimension D=2

Space filling Peano curveFractal dimension D=2


Some results of SLEκ

Loop erasedrandom walkκ = 2, D=5/4

Percolationdomainκ = 6, D=7/4

Ising domainrandom walkκ = 3, D=11/8

Self avoidingrandom walkκ = 8/3, D=4/3


Summary and conclusionSLE is only 5-6 years old – but moving fastSLE represents qualitative progress in 2D critical phenomena in thescaling limitSLE is mainly driven by mathematicians – now the theoreticalphysicists are catching upSLE demonstrates the power of analysis when it worksSLE provides geometrical understanding of conformal field theoryThe SLE parameter κ delimits universality classes in 2DSLE ideas have already been applied to 2D turbulence and spinglassesThere is surely more to come

Thank you for your attention

Stochastic Loewner Evolution - AUusers-phys.au.dk/fogedby/statphysII/notes/SLE2.pdf · Padova April...

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