Conformally Invariant Random Shapes and Schramm-Loewner ...from v 0 and stopped when it leaves D....

Conformally Invariant Random Shapesand Schramm-Loewner Evolution

Dapeng Zhan

Sloan Lecture

Dapeng Zhan Conformally Invariant Random Shapes

Overview

In this talk, I will discuss some two-dimensional lattice models.These models generate random shapes. When the mesh of thelattice tends to 0, the random shape tends to a random fractal setcalled the scaling limit. And the distribution of the scaling limit isinvariant under conformal maps.

These phenomena have been observed by Statistical Physicists fora long time. But for most cases, the rigorous proof is missing, andlittle is known about the scaling limit.

The situation was changed since Oded Schramm introduced theSchramm-Loewner evolution (SLE) in 1999. SLE with differentparameters have been identified as the scaling limits of a numberof lattice models.


Lattices

We work on regular lattices, for example, the square lattice and theequilateral triangular lattice. We often restrict the lattice inside adomain.

The main tools that will be used are Probability Theory andComplex Analysis.


Random Shape

A random shape is a set valued random variable. Recall that for areal valued random variable X , its property depends on itsdistribution: the values of P [a < X < b] for any a < b ∈ R.

For a random set S in R2, the distribution of S is the collection ofvalues: P [S ∩V = ∅], where V is any open set in R2. We say thattwo random sets S1 and S2 have the same distribution ifP [S1 ∩ V = ∅] = P [S2 ∩ V = ∅] for any open set V ⊆ R2.


Conformal Map

We identify the plane R2 with the set of Complex numbers: C.The conformal maps in this talk are bijective holomorphic(analytic) maps. Conformal maps preserve angles.


Conformal Map

Theorem (Riemann’s Mapping Theorem)

Let D1 and D2 be two simply connected domain (other than C).Then there is f that maps D1 conformally onto D2. Moreover, wemay require that f (a1) = a2 and f (b1) = b2 for prescribed pointsa1, a2, b1, b2 in the following two cases.

I aj ∈ ∂Dj and bj ∈ Dj , j = 1, 2.

I aj ∈ ∂Dj , bj ∈ ∂Dj , and aj 6= bj , j = 1, 2.

A random set is called conformally invariant if its distribution ispreserved under conformal maps.


Random Walk

We starts with a simple lattice model: random walk.

A random walk on a square lattice is a random lattice pathX = (X0,X1,X2, . . . ), which is constructed inductively. First wespecify an initial vertex: v0, and let X0 = v0. When X0, . . . ,Xn areknown, the position of the next vertex Xn+1 has 4 possibilities: thefour neighbor vertices of Xn. Each possibility occurs with equalprobability: 1

4 .


Random Walk

The following picture shows the image of a random walk on asquare lattice.


Brownian Motion

It is well known that random walk on regular lattices converge toplanar Brownian motion when the mesh tends to 0.


Brownian Motion

A planar Brownian motion is a random function from [0,∞) intothe plane whose two components are linearly independent 1-dimBrownian motions.

An 1-dim Brownian motion B(t), 0 ≤ t <∞, is a random realvalued continuous function with B(0) = 0, and has independentincrements with distribution B(t)− B(s) ∼ N(0, t − s) fort > s ≥ 0, where N(µ, σ2) denotes the normal distribution withmean µ and variance σ2.


Brownian Motion

A less well-known fact is that planar Brownian motions areconformally invariant.

Theorem [Levy, 1948]

Suppose that f maps D1 conformally onto D2, z1 ∈ D1, and z2 =f (z1) ∈ D2. For j = 1, 2, let Bj(t) be a planar Brownian motionstarted from zj , and let Tj be the first t such that Bj(t) ∈ ∂Dj .Let Sj = Bj [0,Tj ], i.e., the image of the initial part of Bj before itleaves Dj . Then S2 has the same distribution as f (S1).


Loop-erased Random Walk

Gregory Lawler introduced the so-called loop-erased random walk(LERW). It is defined by erasing loops on a finite part of a simplerandom walk. The loop-erasure is defined for any finite latticepath. Let X be a finite path. We do the following

(i) Travel along X . At the first time a vertex is visited twice, weget the first loop on X .

(ii) Erase the first loop from X . Find the first loop on theremaining path, if there is any, and erase it.

(iii) Continue until all loops are erased.

(iv) The remaining simple path is called the loop-erasure of X .









Suppose that D is a simply connected domain, and z0 ∈ D.Consider a square lattice with mesh size δ. Let v0 be a vertex onthe lattice that is closest to z0. Let X be a random walk startedfrom v0 and stopped when it leaves D. The loop-erasure of X iscalled the LERW in D from v0 to ∂D.

It was conjectured that

1. As δ → 0, the image of Y tends to a random simple curve inD connecting z0 with ∂D.

2. Such scaling limit is conformally invariant.



The picture below is a LERW inside a disc.



Theorem [Lawler-Schramm-Werner, 2002]

As the mesh tends to 0, the above LERW converges to a radialSLE(2) curve in D, which is a random simple curve connecting z0

with ∂D, and satisfies conformal invariance.

Wendelin Werner received the Fields medal in 2006 for his work onSchramm-Loewner evolution and the geometry of planar Brownianmotion: he together with Lawler and Schramm used SLE as a toolto prove the conjecture of Mandelbrot that the boundary of planarBrownian motion has Hausdorf dimension 4/3.


Schramm-Loewner Evolution

Schramm’s SLE process generates a non-self-crossing randomcurve in a simply connected domain, which satisfies conformalinvariance. In the picture below, the left curve is self-crossing, andthe right curve is non-simple but non-self-crossing.



There are mainly two versions of SLE: chordal SLE and radial SLE.They both grow in a simply connected domain and start from aboundary point. Chordal SLE ends at another boundary point.Radial SLE ends at an interior point. They both satisfy conformalinvariance. Their geometric properties depend on a parameterκ > 0. We use SLE(κ) to emphasize this parameter.

Theorem [Rohde-Schramm, 2001]

A chordal or radial SLE(κ) curve is simple when κ ≤ 4, non-simple(and non-self-crossing) when κ > 4, space-filling when κ ≥ 8.



Theorem [Beffara, 2008]

The Hausdorff dimension of a chordal or radial SLE(κ) curve ismin{1 + κ

8 , 2}.

The picture below illustrates a part of an SLE(κ) curve in the threedifferent cases: κ ≤ 4, 4 < κ < 8, and κ ≥ 8.


Chordal SLE

The picture below is a complete chordal SLE curve for κ ≤ 4.


Radial SLE

The picture below is a complete radial SLE curve for κ ≤ 4.



The result on the convergence of LERW to radial SLE(2) wasgeneralized.

Theorem [Z, 2006]

Chordal SLE(2) is also the scaling limit of some LERW in a simplyconnected domain. More generally, the LERW in a finitely connectedplane domain also has a scaling limit, which is some SLE(2)-typerandom curve, and satisfies conformal invariance.


Scaling Limits

SLE(2) is not the only kind of SLE that is interesting. BesidesLERW, the following lattice models also converge to SLE.

1. Critical site percolation on a triangular lattice has an interfacewhich converges to chordal SLE(6), Smirnov, 2000.

2. Uniform spanning tree Peano curve converges to chordalSLE(8), Lawler-Schramm-Werner, 2002.

3. Gaussian free field contour line converges to chordal SLE(4),Sheffield-Schramm, 2006.

4. Two types of critical Ising models interfaces converge tochordal SLE(3) and SLE(16/3), Smirnov, 2006 & 2009.

Stanislav Smirnov received the Fields medal in 2010 for his work onproving the convergence of critical percolation and Ising models.


Percolation

In mathematics, percolation theory studies the connectivity in arandom environment. Here we discuss the critical site percolationon a triangular lattice. Use two colors, say yellow and green, tocolor the vertices of a triangle lattice, such that each vertex iscolored independently with probability 1/2 to be yellow andprobability 1/2 to be green. Then we want to study the behaviorsof the yellow clusters and green clusters.

For a triangular lattice, we may draw a hexagonal lattice such thatevery vertex of the triangular lattice is the center of a face of thehexagonal lattice, and vice versa. The site percolation on atriangular lattice is equivalent to the face percolation on ahexagonal lattice.


Percolation


Percolation

To relate percolation with SLE, we consider a simply connecteddomain D with two marked boundary points a, b, and a hexagonallattice with small mesh. Color all hexagon faces contained in Dindependently yellow or green with equal probability.

The boundary points a and b divide ∂D into two arcs. We assign aboundary condition to this percolation by adding a coat of hexagonfaces to the above percolation, and coloring these faces such thatthe faces on one arc are all green and the faces on the other arcare all yellow. Then we can observe an interface curve connectingthe two marked points.


Percolation


Percolation


Percolation


Percolation


Percolation


Percolation


Percolation


Percolation

Theorem [Smirnov, 2000]

The above interface curve converges to chordal SLE(6) in the do-main as the mesh tends to 0. The result does not depend on theshape of the domain or the choices of the marked boundary points.So the scaling limit of the interface is conformally invariant.


Cardy’s formula

Let D be a simply connected domain with 4 boundary pointsa, b, c, d , which divides ∂D into 4 arcs. Consider a face percolation(colored yellow or green) on the hexagonal lattice restricted in D.

Question: What is the limit probability of the event that there is ayellow crossing from ad to bc as the mesh tends to 0? The aboveevent is complement to that there is a green crossing from ab tocd . John Cardy conjectured that the limit exists, is conformallyinvariant, and has a nice expression when the domain is anequilateral triangle with vertices a, b, c , in which case, p = |ad |

|ac| .This conjecture is called the Cardy’s formula.


Cardy’s formula

a

b

c d


Cardy’s formula

For the percolation model we just considered. If there is a yellowhorizontal crossing, the interface curve started from the lower leftcorner must reach the right side before the upper side. If there is agreen vertical crossing, the interface curve must reach the upperside before the right side.

So the limit value Cardy conjectured is the probability that thechordal SLE(6) curve visits the right side before the upper side,which is conformally invariant, and can be easily computed.

In fact, Smirnov first proved Cardy’s formula, and used this toprove the convergence of the interface.


Cardy’s formula

In the picture below, there is a green vertical crossing. Theinterface curve reaches the upper side before the right side.


Uniform Spanning Tree

In graph theory, a tree is a connected graph with no cycles. Let Gbe a connected graph. A spanning tree of G is a subgraph of Gwhich is a tree and contains all vertices of G . Suppose G is finite.Then the number of spanning trees of G is finite. A uniformspanning tree on G is a random spanning tree chosen among allthe possible spanning trees of G with equal probability.

To relate the uniform spanning tree with SLE, we consider a simplyconnected domain D with two marked boundary points a, b.Consider a square lattice with small mesh. Let G be the restrictionof the lattice in D.



Let I be one boundary arc bounded by a, b. Let EI be all edges ofG that lie on I . Let T be a random spanning tree chosen amongall the possible spanning trees of G which contain all edges on EI

with equal probability. The additional requirement can be viewedas a boundary condition.

Given T , we may construct a dual tree T †, which is a uniformspanning tree on the dual graph G †. Then there is an interfacecurve which connects a and b, and separates T from T †. Suchcurve visits every square face of G and G †.



a

b

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b



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b



a

b

a

b



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b

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Theorem [Lawler-Schramm-Werner, 2002]

The Peano curve of the uniform spanning tree with the above bound-ary condition converges to the chordal SLE(8) curve as the meshtends to 0. So the scaling limit of the uniform spanning tree Peanoncurve is conformally invariant.


Definition of SLE

For the definition of chordal SLE, we use the chordal Loewnerequation. Let ξ be a real valued continuous function on [0,∞).Let z ∈ C. The chordal Loewner differential equation driven by ξis the equation

∂

∂tg(t, z) =

2

g(t, z)− ξ(t), g(0, z) = z .

For a fixed z , this is an ordinary differential equation. As tincreases, the definition domain of g(t, ·) decreases. Arguing onthe inverse flow, one see that g(t, ·) is always injective. Since thekernel functions in the equation: z 7→ 2

z−ξ(t) are analytic, g(t, ·) is

also analytic. Thus, g(t, ·) is conformal for every t.


Definition of SLE

Let H denote the upper half plane {z ∈ C : Im z > 0}. Using theinverse flow, one can actually show that g(t, ·)−1 is well defined onH. For the definition of chordal SLE(κ), we let the driving functionξ(t) be

√κB(t), where B(t) is a one-dim Brownian motion and

κ > 0, and let

β(t) = limH3z→ξ(t)

g(t, ·)−1(z), t ∈ [0,∞).

Rohde and Schramm proved that

1. the limit exists for all t ≥ 0;

2. β(t) is a continuous curve on H ∪ R;

3. β(0) = ξ(0) = 0 and limt→∞ β(t) =∞.

4. the geometric property of β depends on the value of κ.


Definition of SLE

The curve β is called a standard chordal SLE(κ). It grows in H,starts from 0 and ends at ∞. Let D be any simply connecteddomain with two boundary points a and b. Let f map Hconformally onto D such that f (0) = a and f (∞) = b. Then f (β)is called a chordal SLE(κ) curve in D from a to b.

Radial SLE is defined using radial Loewner equation. It is firstdefined on the unit disc {z ∈ C : |z | < 1} and then extended togeneral simply connected domains using conformal maps. Thecurve starts from a boundary point and ends at an interior point.

Both kinds of SLE have conformal invariance by definition. Thecomputation on SLE uses Ito’s Stochastic Analysis.


Domain Markov Property

Both chordal SLE and radial SLE satisfy “Domain MarkovProperty” (DMP). This follows from the fact that the drivingfunction

√κB(t) has i.i.d. increments.

Let β be a chordal or radial SLEκ curve in D from a to b. SupposeT is a stopping time for β. The DMP states that: given theinformation of the curve β[0,T ], the rest of β has the distributionof a chordal or radial SLE(κ) in the remaining domain, say Dβ,T

from β(T ) to b. More specifically, if f maps Dβ,T conformallyonto D, fixes b and takes β(T ) to a, then f (βt≥T ) has the samedistribution as the original β.



A stopping time for β is a random time with the property that, ifone travels along β from a to b, when the time T arrives heimmediately knows. Suppose one has memory of the past, but cannot see the future. For example, the first time that β visits a set Sis a stopping time. But the last time that β visits such S is not astopping time because the traveler does not know whether he willvisit S again.

The remaining domain at time T is Dβ,T := D \ β[0,T ] in thecase κ ≤ 4 or κ ≥ 8. If κ ∈ (4, 8), D \ β[0,T ] is not connected,and in this case we let Dβ,T be the connected component ofD \ β[0,T ] whose closure contains b.



The picture below shows the case κ ∈ (4, 8), whereDβ,T = D−(blue line+yellow area).



If β is a non-self-crossing random curve satisfying conformalinvariance and DMP, then it must be a chordal or radial SLE withsome parameter κ > 0. If a lattice model satisfies DMP at thediscrete level, and has a conformally invariant scaling limit, thenthe limit must be some SLE.

In fact, the lattice models which have been proved to converge toSLE all satisfy DMP at the discrete level. We may use thepercolation model to illustrate this.



Let X = (X0,X1, . . . ) be the interface path. We observe that allhexagons on its left are yellow, and all hexagons on its right aregreen. Stop the path at time T . Suppose that we know theinformation of (X0, . . . ,XT ). Then we know the colors of allhexagons which share a side with this path. Remove thesehexagons from D, and call the new domain DT .

The color pattern inside DT is still a percolation, and the boundarycondition for DT is similar to the one for D: yellow on one arc,green on the other arc. So conditioned on (X0, . . . ,XT ), theremaining path is still a percolation interface.



In the picture below, the interface curve is stopped at time T . Thehexagons with holes have determined colors at that time.

T


Reversibility

The curves in the various lattice models which converge to chordalSLE all satisfy reversibility, which means that its two end pointsplay the same role. Thus, chordal SLE(κ) satisfies reversibility forκ = 2, 3, 4, 6, 16/3, 8, i.e., the time-reversal of a chordal SLE(κ)curve is a chordal SLE(κ) curve in the same domain. Thereversibility property is not obvious from the definition or DMP.

It was conjectured that chordal SLE(κ) satisfies reversibility forκ ≤ 8. This is not true for κ > 8.


Reversibility

Theorem [Z, 2007]

For κ ∈ (0, 4], chordal SLE(κ) satisfies reversibility.

The proof does not use lattice models. The main idea is to growtwo SLE curves in the same domain towards each other, andcouple the two curves such that they commute with each otherand eventually overlap.

The reversibility in the case κ ∈ (4, 8) is still open.


Reversibility

Radial SLE can not satisfy reversibility because its two end pointsare topologically different. But one may use radial SLE to definethe whole-plane SLE, which is a random curve growing in theextended complex plane C = C ∪ {∞} from one interior point toanother interior point.

The whole-plane SLE(κ) curve β from a to b satisfies that, for anystopping time T , given the information on β≤T , the rest of thecurve: β≥T is a radial SLE(κ) in the remaining (simply connected)domain from β(T ) to b. This is similar to the DMP for radial SLE.


Reversibility

Theorem [Z, 2010]

For κ ∈ (0, 4], whole-plane SLE(κ) satisfies reversibility.

The proof uses several tools:

1. the stochastic coupling technique used before;

2. the definition of SLE in doubly connected domains;

3. Feynman-Kac formula, used to solve a PDE.


Recent Development: Conformal Loop Ensemble

A family of new random models called conformal loop ensemble(CLE) were defined and studied recently([Sheffield-Werner, 2010]).They are expected to be the scaling limit of the boundary of theclusters in various lattice models, such as critical percolation,Gaussian free field, and critical Ising models.

CLE depends on a parameter κ ∈ (8/3, 8), and its definition usesSLE(κ) with the same parameter. An CLE(κ) is a random shapecomposed of infinitely many non-self-crossing loops. Forκ ∈ (8/3, 4], these are all simple loops and are pairwise disjoint; forκ ∈ (4, 8), the loops are not simple and may touch each other.


Recent Development: Conformal Loop Ensemble

In the picture below, other than the interface curve (red), theboundary between the yellow clusters and green clusters is a unionof loops, which is expected to converge to CLE(6).


Recent Development: Natural Parametrization

The lattice paths which converges to SLE have natural latticelength. If the mesh is δ, and the total number of steps is N, thenthe total length is Nδ. As δ → 0, the path tends to some SLE(κ),which has Hausdorff dimension d = min{1 + κ

8 , 2}. So N ∼ δ−d ,and the total length ∼ δ1−d . Rescale the length of the lattice pathby a factor δd−1 so that the order of magnitude of the length doesnot change as δ → 0.

It is expected that as δ → 0, this rescaled length converges to ameasure supported by the SLE(κ) curve. The limit measure, ifexists, is called the natural parametrization of SLE(κ).


Recent Development: Natural Parametrization

If the natural parametrization of SLE(κ) exists, it must satisfysome nice properties, such as conformal covariance and DomainMarkov Property. Lawler-Sheffield and Lawler-Zhou proved thatthe natural parametrization of SLE(κ) with the desired propertiesexists and is unique (up to a linear factor) for every κ ∈ (0, 8).

No results have been proven about the convergence of the naturallength of the lattice path to the natural parametrization of SLE.


Thank you!


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