Conformally invariant

scaling limits:

Brownian motion, percolation,

and loop-erased random walk

Oded SchrammMicrosoft Research

Weizmann Institute of Science (on leave)

Plan

1. Brownian motion

2. Loop-erased random walk

3. SLE

4. Percolation

5. Uniform spanning trees (UST)

6. UST Peano curve

7. Self-avoiding walk

1

One dimensional Brownian motion

A simple way to describe Brownian motion is as ascaling limit of simple random walk.

Let S(0) = 0 and given S(1); S(2); : : : ; S(n) let S(n+1) := S(n)� 1 with probability 1=2 each. Then S(n)is simple random walk in Z. For Æ > 0 let

SÆ(t) := Æ S(bt Æ�2c) ; t > 0 :

As Æ # 0, SÆ converges to Brownian motion B(t).

One-dimensional Brownian motion is a randomcontinuous path B : [0;1) ! R . It has the Markovproperty: given B(t), the past, Bj[0;t] and the future,Bj[t;1), are independent.

2

Two dimensional Brownian motion

and conformal invariance

Two dimensional Brownian motion is obtained bycollecting together two independent 1-dimensionalBMs, B(t) := (B1(t); B2(t)). It is also the limitof simple random walk on the square grid Z

2.

It turns out that 2D Brownian motion has rotationalsymmetry.

As an unparameterized path, it even has conformalsymmetry.

f

3

Special points on the Brownian path

The Brownian path has some special points on it:outer boundary points, cut points and frontier points,for example. It is interesting to study the sizes of thesesets.

4

Loop-erased random walk

Consider a bounded domain D in the plane. Supposethat 0 2 D, and consider simple random walk S on Z

2

started from 0 and stopped when it exits D.

Let LE(S) be the path obtained by erasing loopsfrom S as they are created. This is the loop-erasedrandom walk (LERW). It was invented by Lawler (as asubstitute for the self-avoiding walk).

One reason for the signi�cance of LERW is that thepaths in the uniform spanning tree (UST) are LERW.

5

Conformal invariance of LERW

Theorem (Lawler-S-Werner). The scaling limit ofloop-erased random walk exists and is conformallyinvariant.

Scaling limit means that we take the limit as the meshof the grid re�nes.

Conformal invariance means the following. Supposethat f : D ! D0 is a conformal map between simply-connected domains, with 0 2 D;D0 and f(0) = 0.Let X be the scaling limit of LERW from 0 in D,and let X 0 be the scaling limit of LERW from 0 inD0. Then f(X) has the same distribution as X, asunparameterized paths.

In fact, we show that the limit is SLE2, stochasticLoewner evolution with parameter 2.

6

LERW as a Markov chain on domains

Consider the LERW from 0 to @D. Let � be asimple path in D with one endpoint in @D, and let qbe the other endpoint. It is a combinatorial identitythat conditioned on � � , the arc �� has the samedistribution as LERW from 0 to @D [ � conditionedto hit @D [ � at q.

0�

q

7

Markov process on conformal maps

Assume conformal invariance of the scaling limit. Thenwe may take the domain to be the unit disk.

Grow the LERW scaling limit from the boundary, andstop when it has diameter �, say.

Although the path may be complicated, we can simplifyby mapping conformally back to the unit disk.

Iterating this gives the conformal map from thecomplement of progressively larger pieces of the pathas the composition of many maps close to the identity,which are independent and identically distributed(except for rotation).

If we take many iterations with small slits, the shapesdo not matter. All that matters in the limit are therelative rotation and \size" as measured by capacity.

8

The setup for Loewner's theorem

The latter statement is a consequence of Loewner'stheorem.

Take U slitted by a path �.

Parameterize � by capacity. This means that �(0) 2@U and the conformal maps

gt : U n �[0; t]! U

normalized by gt(0) = 0, g0t(0) > 0, satisfy also

g0t(0) = et:

9

Loewner's theorem

Loewner's theorem states that in this setting the mapsgt satisfy the ODE

@

@tgt(z) = gt(z)

gt(z) + �(t)

gt(z)� �(t)(1)

where �(t) = gt(�(t)) 2 @U is the image of the tip�(t) under the uniformizing map.

Obviously, we also have the initial condition

g0(z) = z : (2)

10

LERW scaling limit

For the LERW scaling limit curve, the path � :[0;1) ! @U is random. The path �̂(t) := arg �(t)has stationary independent increments, is continuous,and it follows that it must be Brownian motion withtime scaled by some factor.

De�nition (S). (Radial) SLE� is the process obtainedby solving Loewner's di�erential equation (1) with

�(t) := exp(i B(�t))

and the initial condition g0(z) = z.

It can be thought of as a 1-parameter family gt ofconformal maps. But, in fact, we are interested in thehull Kt, the complement of the domain of de�nition ofgt; gt : U nKt ! U .

Theorem (Lawler-S-Werner). The scaling limit ofLERW in U is equal to (radial) SLE2.

We prove this without assuming conformal invariance.

11

Phases of SLE

The SLE trace is the path t 7! g�1t (�(t)).

Theorem (Rohde-S). For all � > 0, � 6= 8, theSLE� trace is a.s. a continuous path. It is a simplepath i� � 6 4. It is space �lling i� � > 8.

Continuity is nontrivial, since it is not a priori clearthat g�1

t extends continuously to the boundary.

� 2 [0; 4] � 2 (4; 8) � 2 [8;1)

In the phase � 2 (4; 8), the SLE path makes loops\swallowing" parts of the domain. However, it nevercrosses itself.

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Percolation

Here is one of several models for percolation.

Fix some p 2 [0; 1]. In Bernoulli(p) percolation,each hexagon is white (open) with probability p,independently. The connected components of thewhite regions are studied.

Various similar models include bond p-percolation onZd.

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Critical Percolation

There is some number pc 2 (0; 1) such that there isan in�nite component with probability 1 if p > pc andwith probability 0 if p < pc.

The large-scale behaviour changes drastically when pincreases past pc. This is perhaps the simplest modelfor a phase transition.

Theorem (Kesten 1980). In the above percolationmodel pc = 1=2.

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Scaling

We are really more interested in large-scale propertiesof percolation. In other words, we would like tounderstand the limiting behaviour of percolation as themesh tends to zero.

This is completely uninteresting unless p = pc or p !pc.

At p = pc, the scaling limit is a natural mathematicalobject, displaying, universality (conjecturally), rotationinvariance, and conformal invariance.

Special to two dimensions.

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Conformal invariance of percolation

Theorem (Smirnov 2001). The scaling limit of thispercolation model exists and is conformally invariant.

This is not a precise statement, for we have not saidin what sense the limit is taken.

Central example: crossing probabilities.

One possible sense is as follows: Let F be the setof all compact connected subsets of the set of whitehexagons inside the domain D. Then percolationmay be thought of as the probability measure whichis the distribution of F. As the mesh goes to zero,these measures tend (weakly) to a limiting probabilitymeasure.

Lacking: a proof for other percolation models, forexample, Z2 bond percolation.

16

Critical percolation boundary path

In the �gure, each of the hexagons is colored blackwith probability 1=2, independently, except that thehexagons intersecting the positive real ray are all white,and the hexagons intersecting the negative real ray areall black. There is a boundary path �, passing through0 and separating the black and the white regionsadjacent to 0. The intersection of � with the upperhalf plane H , is a random path in H connecting theboundary points 0 and 1.

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Critical percolation and SLE

A corollary of Smirnov's theorem is.

Theorem. The scaling limit of the percolationboundary path exists, and is equal to chordal SLE6.

Chordal SLE is essentially the same as radial SLE, butinstead of growing from the boundary to an interiorpoint, it grows from one boundary point to anotherboundary point.

The de�nition is the same, except that �(t) := B(� t)is now BM on R and the di�erential equation is

@

@tgt(z) =

2

gt(z)� �(t); g0(z) = z;

for z in the upper half plane H .

Corollary (LSW). The probability that in the abovepercolation model the origin is connected via blackhexagons to distance R decays like R�5=48+o(1) asR!1.

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The Brownian motion boundary

The outer boundary of BM can be described as anSLE8=3-like process, but where the 1-dimensional BMhas a position dependent drift. Easier to understand isthe relation with SLE6:

Theorem (LSW). Let B be BM in H starting at 0and re ected at an angle of �=3 o� [0;1) and at anangle of 2�=3 o� (�1; 0], stopped on hitting the unitcircle. Let X be the union of the image of B andall bounded components of H n B. Let KT be thehull of chordal SLE6 at the �rst time T such that KT

intersects the unit circle. Then the distribution of KT

is the same as that of X.

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BM boundary and SLE6 boundary

20

Consequences for the BM boundary

Theorem (LSW). With probability 1, the Hausdor�dimension of the outer boundary of 2D BM is 4=3, theset of cut points has Hausdor� dimension 3=4, and theset of pioneer points has Hausdor� dimension 7=4.

Analogous results for the outer boundaries of scalinglimit of percolation clusters...

21

Uniform spanning trees (UST)

Consider a random-uniform spanning tree of an n� nsquare in the grid Z

2.

22

LERW in UST

If you �x two vertices a; b in a �nite graph G, then theUST path joining them is LERW, from a to b.

In fact, the UST can be built by repeatedly takingLERW (Aldous-Broder, Wilson).

Corollary (LSW). The scaling limit of the UST inD \ Z

2 is conformally invariant.

23

The Peano curve associated with the

UST

The complement of the UST in the plane is anotherUST (on a dual grid). Between the UST and its dualwinds the Peano path.

Theorem (LSW). The scaling limit of the UST Peanocurve is conformally invariant.

24

The UST Peano curve and SLE8

Take a domain D with its boundary partitioned intotwo arcs @D = A1 [A2. Consider the UST in D withA1 wired and A2 free.

Theorem (LSW). The scaling limit of the Peano curvefor this UST is the image of chordal SLE8 under theconformal map from H to D taking 0 and 1 to thetwo points in A1 \A2.

25

Problem: the self avoiding walk

The uniform measure on self avoiding walks of lengthn has been notoriously hard to study (as n ! 1). Itis even hard to simulate precisely.

Let Xn be random-uniform among all simple paths inZ2 \ H of length n which start at 0. Let X1 be the

distributional limit of Xn, as n ! 1. Let X be thescaling limit of X1: that is, the distributional limit ofÆ X1 as Æ ! 0.

Conjecture (LSW). X is chordal SLE8=3.

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