Percolation, Brownian Motion and

SLE

Oded SchrammThe Weizmann Institute of Science

and Microsoft Research

Plan

1. This talk:

(a) Percolation and criticality(b) Conformal invariance Brownian motion (BM) and

percolation(c) Loop-erased random walk (LERW)(d) SLE2 is (conj) the scaling limit of LERW(e) SLE6 is the scaling limit of percolation boundary

curves(f) The two-dimensional BM exponents, and the

dimension of the BM boundary

2. Next talk: Other processes converging to SLE,properties of SLE, and how to compute with SLE

3. Last talk: The determination of the BM exponents

1

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.

2

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.

3

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 !@c.

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

Special to two dimensions.

4

Simple random walk and Brownian

motion

Consider simple random walk on a �ne square grid,which starts at 0 and stops when you hit the boundaryof some speci�ed domain.

When the mesh tends to zero (and time is scaledappropriately) the simple random walk converges toBrownian motion (BM).

5

Conformal invariance of BM

BM has rotational and even conformal invariance, ifone forgets the time parameterization.

6

Conformal invariance of percolation

Theorem (Smirnov 2001). The scaling limit ofpercolation exists and is conformally invariant

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

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.

7

Cardy's formula (Carleson's version)

8x 2 [0; 1],

Px

1

xmesh 0

Cardy (who is a physicist) did not prove this formula.The proof of this formula is central to Smirnov's work.

8

Loop-erased random walk

Consider a bounded domain D in the plane, and letÆZ2 be the square grid of mesh Æ.

Suppose that 0 2 D, and consider simple random walkS on ÆZ2 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.

9

Scaling limit of LERW

Conjecture (folklore). The limit of LERW as Æ ! 0exists, and is conformally invariant.

This means that there is a weak limit of the probabilitymeasure which is the law of LERW (as a compact set,say).

Support for the conjecture comes from simulations,analogies, and some properties of LERW which havebeen proved to be conformally invariant by Kenyon.

Theorem (S). Every subsequential limit is a simplepath a.s.

10

Determining the scaling limit

Let be the scaling limit of LERW from 0 to @U .

Reverse , so that it starts at @U and goes to 0. Forevery t, consider the Riemann map

gt : U n [0; t]! U

normalized by gt(0) = 0 and g0t(0) > 0. Weparameterize so that g0t(0) = exp t.

This is the conformally natural parameterization of .

0

(t)

0

�(t)

Let �(t) := gt( (t)).

11

Determining the scaling limit (cont)

Theorem (S). Assuming the existence and conformalinvariance of the scaling limit of LERW, �(t) has thesame law as B(2t), where B(t) is Brownian motion on@U started from a random-uniform point.

In fact, one can reconstruct (t) from �(t). This isthe content of Loewner's theorem. Using Loewner'stheorem and the above, we get,

12

The scaling limit of LERW

Corollary. Assuming that LERW has a conformallyinvariant scaling limit, the scaling limit of LERW from0 to @U is the path

(t) = g�1t (�(t));

where�(t) = B(2t);

B(t) is BM on @U , and gt is de�ned by Loewner'sequation with parameter �:

@

@tgt(z) = gt(z)

�(t) + gt(z)

�(t)� gt(z); g0(z) = z:

13

Radial SLE

Fix � > 0, let B(t) be BM on @U , and set �(t) :=B(�t). For each z 2 U , let gt = gt(z) be the solutionof the di�erential equation

@

@tgt(z) = gt(z)

�(t) + gt(z)

�(t)� gt(z); g0(z) = z:

Let Dt be the set of points z 2 U such that gs(z) existsand is well de�ned for all s 2 [0; t]. Then gt : Dt ! U

is conformal.

The process (gt; t > 0) is called radial SLE�.

(t) := g�1t (�(t)) is the SLE trace and Kt := U nDt

is the SLE hull.

So the conformal invariance conjecture for LERWimplies that LERW from 0 in U is the same as thetrace of radial SLE2.

14

Chordal SLE

Chordal SLE is essentially the same, but instead ofgrowing from the boundary to an interior point, itgrows from one boundary point to another boundarypoint.

The de�nition is the same, except that B(t) is nowBM on R and the di�ential equation is

@

@tgt(z) =

2

gt(z)� �(t); g0(z) = z; z 2 H :

15

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.

16

Critical percolation and SLE

A corollary of Smirnov's work is.

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

This allows the calculation of properties of percolation.(More next talk.)

17

Brownian intersection exponents

Consider BM in the plane.

The simplest BM exponent is �(1; 1):

R

P[B \B0 = ;] = R��(1;1)+o(1):

18

Signi�cance of the exponents

The exponents encode much information about BMand SRW. For example, the probability that two SRWpaths of n steps each starting from zero will notintersect again decays like n��(1;1)=2 (Burdzy-Lawler).

The dimension of the set of cut points of B[0; 1] is a.s.2� �(1; 1) (Lawler).

19

Determination of the exponents

The values of the exponents �(1; 1; : : : ; 1) havebeen conjectured by Duplantier-Kwon. We prove ageneralization of this:

Theorem (Lawler-S-Werner).

�(n1; n2; : : : ; nk) =

(p24n1 + 1 + : : :+

p24nk + 1� k)2 � 4

48:

Corollary. The Hausdor� dimension of the set of cutpoints of B[0; 1] is a.s. 3=4.

The exponents are determined by showing that theyare the same as the exponents for SLE6 and calculatingthe exponents for SLE.

20

Brownian frontier

Theorem (LSW). The Hausdor� dimension of theouter boundary of B[0; 1] is a.s. 4=3. (As conjecturedby Mandelbrot.)

21

Next time...

Next talk: Other processes converging to SLE,computations with SLE, and properties of SLE.

Last talk: On the BM exponents and theirdetermination via SLE.

22

Plan

1. This talk:

(a) Several Random processes(b) SLE2 as the LERW scaling limit(c) Other processes conjectured to converge to SLE�.(d) Basic properties of SLE(e) Computing with SLE

2. Last talk: More about the BM exponents and theirdetermination

23

Uniform spanning trees (UST)

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

2.

24

Loop-erased random walk

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

25

The de�nition of LERW

The LERW is obtained by performing SRW, andremoving loops as they are created.

In other words, in the loop-erasure of a path , at eachstep you go from a vertex v along the last edge of incident with v. The notion of LERW was introducedby Greg Lawler. The UST relation was �rst discoveredby Aldous-Browder and Pemantle.

26

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.

27

The Big Conjecture

Conjecture. Percolation, UST, LERW and the Peanocurve are conformally invariant in the scaling limit.

Special to 2 dimensions.

Rick Kenyon has shown that some properties of LERWand UST are conformally invariant in the scaling limit.His work is based on the relation with domino tilings.

28

The LERW scaling limit

How does one study the scaling limit of LERW? Theclue is that while the geometry is complicated, theconformal geometry is simple.

29

Fundamental combinatorial property

Consider the LERW from a vertex v to @GÆ. Let� be a simple path in GÆ with one endpoint in @GÆ,and let q be the other endpoint. It is a combinatorialidentity that conditioned on � � , the arc � � hasthe same distribution as LERW from v to @GÆ [ �conditioned to hit @GÆ [ � at q.

v�

q

30

Consider D = U , the unit disk, and let be the scalinglimit of LERW from 0.

Suppose that we know [0; t], what information doesthat give us about the rest of ?

The combinatorial identity implies that the dependenceof [t; t0] on [0; t] is simple conformally: we just needto apply the conformal map taking U n [0; t] onto U .

0

(t)

0

�(t)

31

0

(t)

0

�(t)

What parameterization do we choose for ?

We need some conformally natural parameterization.Take : [0;1]! U so that the Riemann map

gt : U n [0; t]! U

normalized by gt(0) = 0, g0t(0) > 0 satis�es g0t(0) = et.Let �(t) := gt( (t)). Then the above combinatorialidentity for LERW together with conformal invariancetranslate to the Markov property and stationarity for�(t).

32

A process on @U that is stationary, continuous, andhas the Markov property must be Browian motion withtime scaled by some constant. Therefore,

Theorem (S). Assuming the conformal invariance andexistence of the scaling limit of LERW, there is aconstant � > 0 such that �(t) has the same law asB(�t), where B(t) is Brownian motion on @U startedfrom a random-uniform point.

In fact, � = 2 in this case.

In order to determine the time scaling constant 2, onehas to do some calculation. It is determined by theasymptotics of the variance of the winding numberin an annulus with radii � and 1 about 0 and thedetermination by Kenyon of the corresponding variancefor LERW.

33

Cororllary (S). Assuming that LERW has aconformally invariant scaling limit, the scaling limitis radial SLE2.

Reminder: this means that the scaling limit path isgiven by (t) = g�1t (�(t)), where �(t) = B(2t), B(t)is BM on @U starting from a random-uniform point,and gt is the solution of the equation

@

@tgt(z) = gt(z)

�(t) + gt(z)

�(t)� gt(z); g0(z) = z:

34

Percolation is similar

Note that the analogue to the combinatorial propertyfor LERW holds for percolation.

The fact that the path joins boundary points meansthat chordal rather than radial SLE is appropriate.

The fact that � = 6 for percolation follows bycomputing Cardy's formula (for a square, the crossingprobability is 1=2, by duality).

35

The UST Peano and SLE

Assuming the conformal invariance of the LERWscaling limit, it follows that the UST Peano pathis also conformally invariant, and that a variant of it(to make it start and end in distinct boundary points)is the trace of chordal SLE8.

36

Simple paths

Conjecture. Consider the uniform measure on simplegrid paths from 0 to the boundary of U . The scalinglimit exists and equal to radial SLE8. Similarly, thescaling limit of uniform measure on simple grid pathsjoining two speci�ed boundary points of U is equal tochordal SLE8 (mapped conformally from H onto U ).

37

� = 4

We have seen that SLE6 describes the scaling limitof critical percolation boundary paths, and thatconjecturally, SLE2 and SLE8 are scaling limits ofthe LERW and the UST Peano paths. A processconjectured to converge to SLE4 is Kenyon's domino-di�erence contour:

There are also candidates for various other �.

38

Phases of SLE

Theorem (Rohde-Schramm). For all � > 0, � 6= 8,the SLE� 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�1t 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.

The Hausdor� dimension of the SLE path isconjectured to be 1 + �=8 when � 6 8. We havea proof that the expected number of balls of radius

39

� needed to cover the trace (within a bounded set)grows like ��(1+�=8).

40

Ito's formula

Want to di�erentiate functions of BM with respect tot:

d

dtF (Bt) =?

Even more generally, we may have some process Ytsuch that

dYt = a(t) dt+ b(t) dBt :

About all you need to know about stochastic calculusis Ito's formula:

dF (Yt) = F 0(Yt) dYt + (1=2)F 00(Yt) b(t)2 dt:

41

Calculating with SLE

Many things can be calculated about the models usingthe SLE representation.

Theorem (S). Assuming that critical percolation hasa conformally invariant scaling limit, the probabilitythat there is a percolation cluster in U that intersectsa given arc A � @U of length � and separates 0 fromthe complement of A has limit

1

2� �(2=3)p

� �(1=6)2F1

�1

2;2

3;3

2;� cot2

�

2

�cot

�

2:

A

0

42

Proof (sketch)

We consider chordal SLE� joining the endpoints of A.We map conformally to the upper half plane by a mapwhich takes the endpoints of A to 0 and 1.

Recall the maps gt given by the ODE

@tgt =2

gt � �t:

For every point z, there is a �rst time �z when gt(z)hits the singularity �t. This is also the �rst time t whenthe SLE path (t) = g�1t (�t) separates z from 1.

43

Proof (cont.)

z

We are interested in the probability of the event Qz

that the loop around z which closes at time �z ispositively oriented around z. That event is equivalentto

limt%�z

Re(gt(z)� �t)

Im(gt(z)� �t)= +1;

and the limit is �1 if the loop is negatively oriented.Let h(z) = P[Qz].

44

Proof (cont.)

By the Markov property for BM h(gt(z) � �t) is aMartingale. Consequently, Ito's formula gives

�

2@2xh+rh � @tgt(z) = 0 :

By the scaling property, h(z) depends only on thedirection z=jzj. Since h is a function of one realvariable, the above PDE reduces to an ODE. (We sett = 0.) The ODE can be solved for h.

45

SLE� Summary

� conj process dim magic0 line seg 12 LERW 5=4 Wilson's alg4 domino di�erence 6=4 critical6 percolation boundary 7=4 locality8 UST Peano path 8=4 space �lling

Other values of � 2 [0; 8] probably correspond toboundaries of critical random cluster measures.

46

Plan

1. BM intersection exponents and applications

2. Relation of SLE and BM

3. Computing exponents with SLE

4. Future directions

This talk is about joint work with Greg Lawler andWendelin Werner.

47

Brownian intersection exponents

Consider BM in the plane.

The simplest BM exponent is �(1; 1):

R

P[B \B0 = ;] = R��(1;1)+o(1):

48

Signi�cance of the exponents

The exponents encode much information about BMand SRW. For example, the probability that two SRWpaths of n steps each starting from zero will notintersect again decays like n��(1;1)=2 (Burdzy-Lawler).

The dimension of the set of cut points of B[0; 1] is a.s.2� �(1; 1) (Lawler).

49

Determination of the exponents

The values of the exponents �(1; 1; : : : ; 1) havebeen conjectured by Duplantier-Kwon. We prove ageneralization of this:

Theorem (Lawler-S-Werner).

�(n1; n2; : : : ; nk) =

(p24n1 + 1 + : : :+

p24nk + 1� k)2 � 4

48:

Corollary. The Hausdor� dimension of the set of cutpoints of B[0; 1] is a.s. 3=4.

50

Brownian frontier

Theorem (LSW). The Hausdor� dimension of theouter boundary of B[0; 1] is a.s. 4=3. (As conjecturedby Mandelbrot.)

51

Brownian frontier exponent

It was an earlier result of Lawler that the dimensionof the frontier is equal to 2 � �(2; 0), where �(2; 0) isde�ned as exponent of decay for the probability thattwo independent BM's starting at 1 will not separate0 from 1 before hitting the circle of radius R.

The event de�ning �(2; 0).

52

BM exponents and SLE6

Lawler and Werner had an earlier paper showing thatthe BM exponents are the same as for other processessatisfying certain axioms. Our proof generally followedthat strategy, and the main steps were to show that the(slightly modi�ed) axioms are satis�ed and to calculatethe exponents for SLE.

Now, we have a better understanding of the relationbetween BM and SLE6.

53

BM frontier and SLE6

For domains other than H or U , de�ne SLE by mappingconformally. Then SLE is (trivially) conformallyinvariant.

SLE6 is also local. This means that up to time change,the SLE6 trace does not feel where the boundary ofthe domain is, except when touching it.

Locality easily follows from the convergence ofpercolation to SLE6, however we had to work hardto prove locality, because Smirnov's theorem was notestablished at that time.

Conformal invariance+ Locality) BM frontier

54

Consider radial SLE6 from a small circle �@U to 1.Let be the limit of the trace as �! 0.

Given a bounded domain D � R2, consider the hitting

measure for , that is, the probability measure on @Dwhich is the law of the �rst point of in @D.

Conformal invariance and locality imply that the hittingmeasure for is the same as for BM starting from 0.

Stop and BM when we hit the unit circle @U . LetY be the set of points separated by from @U , andsimilarly YB for the BM.

Claim. Y and YB have the same distribution.

55

Proof of claim

Consider a connected setK � U such thatK\@U 6= ;.The probability that Y \ K 6= ; is the harmonicmeasure of K as a subset of @U [ K, and the samefor YB. Hence, for every such K,

P[Y \K = ;] = P[YB \K = ;]:

This suÆces.

56

Computing exponents for SLE

Do the simplest example of �(1; 1).

In radial SLE, the ODE is

@tgt(z) = gt(z)�(t) + gt(z)

�(t)� gt(z):

The time parameter t satis�es g0t(0) = exp(t).

The distance from 0 to [0; t] is about exp(�t) (withat most an error by a factor of 4).

If we want to measure the probability that anotherBM (or SLE6) from 0 will hit @U without intersectingthe current SLE, then what we want is the harmonicmeasure from 0 of @U in the domain U n [0; t]. Thisis the same as

length(gt(@U )) =

Z@U

jg0t(z)j jdzj :

57

Conditioned on the SLE, the probability that anotherBM will not intersect it is

Z@U

jg0t(z)j jdzj :

So the unconditioned probability is the expectation ofthis quantity, which can be proved to be approximately

Ejg0t(1)j ; t = logR :

58

Estimating g0

t(1)

LetF (�; t) := E[ jg0t(1)j j log �(0) = � ] :

If we are given � in the range [0; s], then the conditionedexpected value of Ejg0t(1)j is

jg0s(1)jF (log(gs(1)=�(s)); t� s) :

This is by the chain rule and the Markov property.

In probabilistic jargon, this means that the aboveexpression is a martingale.

If we di�erentiate with respect to s at s = 0, therecannot be a drift term. Ito's formula then gives aparabolic PDE for F as a function of two variables.

The slowest decaying solution for the PDE (as afunction of t); that is, the heighest eigenfunction,can be guessed. It is just

exp(��t) sin(�=2) ;59

where

�(�) =4 + �+

p(�� 4)2 + 16

16:

When we take � = 6, we obtain the exponent �(1; 1) =�(6) = 5=4.

60

What next?

Open problem:

1. Prove that the LERW is conformally invariant. (Alsogives UST and UST Peano).

2. Prove Smirnov's theorem for other percolationmodels.

3. There's a conjectured duality for SLE, where if��0 = 16 and � > 4 then SLE�0 \describes" theouter boundary of SLE�.

4. SLE8=3 is the BM frontier, in the appropriate sense.

5. Reversibility of the BM frontier.

6. Better understanding of SLE.

7. Derive the percolation exponents in the discretesetting.

61