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

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1 Conformally invariant scaling limits: Brownian motion, percolation, and loop-erased random walk Oded Schramm Microsoft Research Weizmann Institute of Science (on leave)

2 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

3 One dimensional Brownian motion A simple way to describe Brownian motion is as a scaling 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 ffi > 0 let S ffi (t) := ffi S(btffi 2 c) ; t > 0 : As ffi # 0, S ffi converges to Brownian motion B(t). One-dimensional Brownian motion is a random continuous path B : [0; 1)! R. It has the Markov property: given B(t), the past, Bj [0;t] and the future, Bj [t;1), are independent. 2

4 Two dimensional Brownian motion and conformal invariance Two dimensional Brownian motion is obtained by collecting together two independent 1-dimensional BMs, B(t) := (B 1 (t);b 2 (t)). It is also the limit of simple random walk on the square grid Z 2. It turns out that 2D Brownian motion has rotational symmetry. As an unparameterized path, it even has conformal symmetry. f 3

5 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 these sets. 4

6 Loop-erased random walk Consider a bounded domain D in the plane. Suppose that 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 loops from S as they are created. This is the loop-erased random walk (LERW). It was invented by Lawler (as a substitute for the self-avoiding walk). One reason for the significance of LERW is that the paths in the uniform spanning tree (UST) are LERW. 5

7 Conformal invariance of LERW Theorem (Lawler-S-Werner). The scaling limit of loop-erased random walk exists and is conformally invariant. Scaling limit means that we take the limit as the mesh of the grid refines. Conformal invariance means the following. Suppose that f : D! D 0 is a conformal map between simplyconnected domains, with 0 2 D; D 0 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 in D 0. Then f (X) has the same distribution as X, as unparameterized paths. In fact, we show that the limit is SLE 2, stochastic Loewner evolution with parameter 2. 6

8 LERW as a Markov chain on domains Consider the LERW fl from 0 Let ff be a simple path in D with one endpoint and let q be the other endpoint. It is a combinatorial identity that conditioned on ff ρ fl, the arc fl ff has the same distribution as LERW from 0 [ ff conditioned to [ ff at q. 0 fl ff q 7

9 Markov process on conformal maps Assume conformal invariance of the scaling limit. Then we may take the domain to be the unit disk. Grow the LERW scaling limit from the boundary, and stop when it has diameter ffl, say. Although the path may be complicated, we can simplify by mapping conformally back to the unit disk. Iterating this gives the conformal map from the complement of progressively larger pieces of the path as 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 shapes do not matter. All that matters in the limit are the relative rotation and size" as measured by capacity. 8

10 The setup for Loewner's theorem The latter statement is a consequence of Loewner's theorem. Take U slitted by a path ff. Parameterize ff by capacity. This means that ff(0) and the conformal maps g t : U n ff[0;t]! U normalized by g t (0) = 0, g 0 t (0) > 0, satisfy also g 0 t (0) = et : 9

11 Loewner's theorem Loewner's theorem states that in this setting the maps g t satisfy g t(z) = g t (z) g t(z) + ο(t) g t (z) ο(t) (1) where ο(t) = g t (ff(t)) is the image of the tip ff(t) under the uniformizing map. Obviously, we also have the initial condition g 0 (z) = z : (2) 10

12 LERW scaling limit For the LERW scaling limit curve, the path ο : [0; is random. The path ^ο(t) := arg ο(t) has stationary independent increments, is continuous, and it follows that it must be Brownian motion with time scaled by some factor. Definition (S). (Radial) SLE» is the process obtained by solving Loewner's differential equation (1) with ο(t) := exp(ib(»t)) and the initial condition g 0 (z) = z. It can be thought of as a 1-parameter family g t of conformal maps. But, in fact, we are interested in the hull K t, the complement of the domain of definition of g t ; g t : U n K t! U. Theorem (Lawler-S-Werner). The scaling limit of LERW in U is equal to (radial) SLE 2. We prove this without assuming conformal invariance. 11

13 Phases of SLE The SLE trace is the path t 7! g t 1 (ο(t)). Theorem (Rohde-S). For all» > 0,» 6= 8, the SLE» trace is a.s. a continuous path. It is a simple path iff» 6 4. It is space filling iff» > 8. Continuity is nontrivial, since it is not a priori clear that g t 1 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 never crosses itself. 12

14 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 the white regions are studied. Various similar models include bond p-percolation on Z d. 13

15 Critical Percolation There is some number p c 2 (0; 1) such that there is an infinite component with probability 1 if p > p c and with probability 0 if p < p c. The large-scale behaviour changes drastically when p increases past p c. This is perhaps the simplest model for a phase transition. Theorem (Kesten 1980). In the above percolation model p c = 1=2. 14

16 Scaling We are really more interested in large-scale properties of percolation. In other words, we would like to understand the limiting behaviour of percolation as the mesh tends to zero. This is completely uninteresting unless p = p c or p! p c. At p = p c, the scaling limit is a natural mathematical object, displaying, universality (conjecturally), rotation invariance, and conformal invariance. Special to two dimensions. 15

17 Conformal invariance of percolation Theorem (Smirnov 2001). The scaling limit of this percolation model exists and is conformally invariant. This is not a precise statement, for we have not said in what sense the limit is taken. Central example: crossing probabilities. One possible sense is as follows: Let F be the set of all compact connected subsets of the set of white hexagons inside the domain D. Then percolation may be thought of as the probability measure which is the distribution of F. As the mesh goes to zero, these measures tend (weakly) to a limiting probability measure. Lacking: a proof for other percolation models, for example, Z 2 bond percolation. 16

18 Critical percolation boundary path In the figure, each of the hexagons is colored black with probability 1=2, independently, except that the hexagons intersecting the positive real ray are all white, and the hexagons intersecting the negative real ray are all black. There is a boundary path fi, passing through 0 and separating the black and the white regions adjacent to 0. The intersection of fi with the upper half plane H, is a random path in H connecting the boundary points 0 and 1. 17

19 Critical percolation and SLE A corollary of Smirnov's theorem is. Theorem. The scaling limit of the percolation boundary path exists, and is equal to chordal SLE 6. Chordal SLE is essentially the same as radial SLE, but instead of growing from the boundary to an interior point, it grows from one boundary point to another boundary point. The definition is the same, except that ο(t) := B(»t) is now BM on R and the differential g t(z) = for z in the upper half plane H. 2 g t (z) ο(t) ; g 0(z) = z; Corollary (LSW). The probability that in the above percolation model the origin is connected via black hexagons to distance R decays like R 5=48+o(1) as R! 1. 18

20 The Brownian motion boundary The outer boundary of BM can be described as an SLE 8=3 -like process, but where the 1-dimensional BM has a position dependent drift. Easier to understand is the relation with SLE 6 : Theorem (LSW). Let B be BM in H starting at 0 and reflected at an angle of ß=3 off [0; 1) and at an angle of 2ß=3 off ( 1; 0], stopped on hitting the unit circle. Let X be the union of the image of B and all bounded components of H n B. Let K T be the hull of chordal SLE 6 at the first time T such that K T intersects the unit circle. Then the distribution of K T is the same as that of X. 19

21 BM boundary and SLE 6 boundary 20

22 Consequences for the BM boundary Theorem (LSW). With probability 1, the Hausdorff dimension of the outer boundary of 2D BM is 4=3, the set of cut points has Hausdorff dimension 3=4, and the set of pioneer points has Hausdorff dimension 7=4. Analogous results for the outer boundaries of scaling limit of percolation clusters... 21

23 Uniform spanning trees (UST) Consider a random-uniform spanning tree of an n n square in the grid Z 2. 22

24 LERW in UST If you fix two vertices a; b in a finite graph G, then the UST path joining them is LERW, from a to b. In fact, the UST can be built by repeatedly taking LERW (Aldous-Broder, Wilson). Corollary (LSW). The scaling limit of the UST in D Z 2 is conformally invariant. 23

25 The Peano curve associated with the UST The complement of the UST in the plane is another UST (on a dual grid). Between the UST and its dual winds the Peano path. Theorem (LSW). The scaling limit of the UST Peano curve is conformally invariant. 24

26 The UST Peano curve and SLE 8 Take a domain D with its boundary partitioned into two = A 1 [ A 2. Consider the UST in D with A 1 wired and A 2 free. Theorem (LSW). The scaling limit of the Peano curve for this UST is the image of chordal SLE 8 under the conformal map from H to D taking 0 and 1 to the two points in A 1 A 2. 25

27 Problem: the self avoiding walk The uniform measure on self avoiding walks of length n has been notoriously hard to study (as n! 1). It is even hard to simulate precisely. Let X n be random-uniform among all simple paths in Z 2 H of length n which start at 0. Let X 1 be the distributional limit of X n, as n! 1. Let X be the scaling limit of X 1 : that is, the distributional limit of ffi X 1 as ffi! 0. Conjecture (LSW). X is chordal SLE 8=3. 26

Plan 1. This talk: (a) Percolation and criticality (b) Conformal invariance Brownian motion (BM) and percolation (c) Loop-erased random walk (LERW) (d

Percolation, Brownian Motion and SLE Oded Schramm The Weizmann Institute of Science and Microsoft Research Plan 1. This talk: (a) Percolation and criticality (b) Conformal invariance Brownian motion (BM)