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

Transcription

1 Percolation, Brownian Motion and SLE Oded Schramm The Weizmann Institute of Science and Microsoft Research

2 Plan 1. This talk: (a) Percolation and criticality (b) Conformal invariance Brownian motion (BM) and percolation (c) Loop-erased random walk (LERW) (d) SLE 2 is (conj) the scaling limit of LERW (e) SLE 6 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

3 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. 2

4 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. 3

5 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 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. 4

6 Simple random walk and Brownian motion Consider simple random walk on a fine square grid, which starts at 0 and stops when you hit the boundary of some specified domain. When the mesh tends to zero (and time is scaled appropriately) the simple random walk converges to Brownian motion (BM). 5

7 Conformal invariance of BM BM has rotational and even conformal invariance, if one forgets the time parameterization. 6

8 Conformal invariance of percolation Theorem (Smirnov 2001). The scaling limit of percolation exists and is conformally invariant This is not a precise statement, for we have not said in what sense the limit is taken. 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. 7

9 Cardy's formula (Carleson's version) 8x 2 [0; 1], x P x mesh 0 1 Cardy (who is a physicist) did not prove this formula. The proof of this formula is central to Smirnov's work. 8

10 Loop-erased random walk Consider a bounded domain D in the plane, and let ffiz 2 be the square grid of mesh ffi. Suppose that 0 2 D, and consider simple random walk S on ffiz 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. 9

11 Scaling limit of LERW Conjecture (folklore). The limit of LERW as ffi! 0 exists, and is conformally invariant. This means that there is a weak limit of the probability measure 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 have been proved to be conformally invariant by Kenyon. Theorem (S). Every subsequential limit is a simple path a.s. 10

12 Determining the scaling limit Let fl be the scaling limit of LERW from 0 Reverse fl, so that it starts and goes to 0. For every t, consider the Riemann map g t : U n fl[0;t]! U normalized by g t (0) = 0 and g 0 t (0) > 0. We parameterize fl so that g 0 t (0) = exp t. This is the conformally natural parameterization of fl. fl(t) (t) 0 fl 0 Let (t) := g t (fl(t)). 11

13 Determining the scaling limit (cont) Theorem (S). Assuming the existence and conformal invariance of the scaling limit of LERW, (t) has the same law as B(2t), where B(t) is Brownian motion started from a random-uniform point. In fact, one can reconstruct fl(t) from (t). This is the content of Loewner's theorem. Using Loewner's theorem and the above, we get, 12

14 The scaling limit of LERW Corollary. Assuming that LERW has a conformally invariant scaling limit, the scaling limit of LERW from 0 is the path fl(t) = g 1 t ( (t)); where (t) = B(2t); B(t) is BM and g t equation with parameter : is defined g t(z) = g t (z) (t) + g t(z) (t) g t (z) ; g 0(z) = z: 13

15 Radial SLE Fix» > 0, let B(t) be BM and set (t) := B(»t). For each z 2 U, let g t = g t (z) be the solution of the g t(z) = g t (z) (t) + g t(z) (t) g t (z) ; g 0(z) = z: Let D t be the set of points z 2 U such that g s (z) exists and is well defined for all s 2 [0;t]. Then g t : D t! U is conformal. The process (g t ; t > 0) is called radial SLE». fl(t) := g 1 t ( (t)) is the SLE trace and K t := U n D t is the SLE hull. So the conformal invariance conjecture for LERW implies that LERW from 0 in U is the same as the trace of radial SLE 2. 14

16 Chordal SLE Chordal SLE is essentially the same, 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 B(t) is now BM on R and the diffential g t(z) = 2 g t (z) (t) ; g 0(z) = z; z 2 H : 15

17 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. 16

18 Critical percolation and SLE A corollary of Smirnov's work is. Theorem. The scaling limit of the percolation boundary path exists, and is equal to chordal SLE 6. This allows the calculation of properties of percolation. (More next talk.) 17

19 Brownian intersection exponents Consider BM in the plane. The simplest BM exponent is ο(1; 1): R P[B B 0 = ;] = R ο(1;1)+o(1) : 18

20 Significance of the exponents The exponents encode much information about BM and SRW. For example, the probability that two SRW paths of n steps each starting from zero will not intersect again decays like n ο(1;1)=2 (Burdzy-Lawler). The dimension of the set of cut points of B[0; 1] is a.s. 2 ο(1; 1) (Lawler). 19

21 Determination of the exponents The values of the exponents ο(1; 1;:::;1) have been conjectured by Duplantier-Kwon. We prove a generalization of this: Theorem (Lawler-S-Werner). ο(n 1 ;n 2 ;:::;n k ) = p ( 24n p :::+ 24n k + 1 k) 2 4 : 48 Corollary. The Hausdorff dimension of the set of cut points of B[0; 1] is a.s. 3=4. The exponents are determined by showing that they are the same as the exponents for SLE 6 and calculating the exponents for SLE. 20

22 Brownian frontier Theorem (LSW). The Hausdorff dimension of the outer boundary of B[0; 1] is a.s. 4=3. (As conjectured by Mandelbrot.) 21

23 Next time... Next talk: Other processes converging to SLE, computations with SLE, and properties of SLE. Last talk: On the BM exponents and their determination via SLE. 22

24 Plan 1. This talk: (a) Several Random processes (b) SLE 2 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 their determination 23

25 Uniform spanning trees (UST) Consider a random-uniform spanning tree of an n n square in the grid Z 2. 24

26 Loop-erased random walk If you fix two vertices a; b in a finite graph G, then the UST path joining them is LERW, from a to b. 25

27 The definition of LERW The LERW is obtained by performing SRW, and removing loops as they are created. In other words, in the loop-erasure of a path fl, at each step you go from a vertex v along the last edge of fl incident with v. The notion of LERW was introduced by Greg Lawler. The UST relation was first discovered by Aldous-Browder and Pemantle. 26

28 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. 27

29 The Big Conjecture Conjecture. Percolation, UST, LERW and the Peano curve are conformally invariant in the scaling limit. Special to 2 dimensions. Rick Kenyon has shown that some properties of LERW and UST are conformally invariant in the scaling limit. His work is based on the relation with domino tilings. 28

30 The LERW scaling limit How does one study the scaling limit of LERW? The clue is that while the geometry is complicated, the conformal geometry is simple. 29

31 Fundamental combinatorial property Consider the LERW fl from a vertex v ffi. Let ff be a simple path in G ffi with one endpoint ffi, 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 v ffi [ ff conditioned to ffi [ ff at q. v fl ff q 30

32 Consider D = U, the unit disk, and let fl be the scaling limit of LERW from 0. Suppose that we know fl[0;t], what information does that give us about the rest of fl? The combinatorial identity implies that the dependence of fl[t; t 0 ] on fl[0;t] is simple conformally: we just need to apply the conformal map taking U n fl[0;t] onto U. fl(t) (t) 0 fl 0 31

33 fl(t) (t) 0 fl 0 What parameterization do we choose for fl? We need some conformally natural parameterization. Take fl : [0; 1]! U so that the Riemann map g t : U n fl[0;t]! U normalized by g t (0) = 0, g 0 t (0) > 0 satisfies g0 t (0) = et. Let (t) := g t (fl(t)). Then the above combinatorial identity for LERW together with conformal invariance translate to the Markov property and stationarity for (t). 32

34 A process that is stationary, continuous, and has the Markov property must be Browian motion with time scaled by some constant. Therefore, Theorem (S). Assuming the conformal invariance and existence of the scaling limit of LERW, there is a constant» > 0 such that (t) has the same law as B(»t), where B(t) is Brownian motion started from a random-uniform point. In fact,» = 2 in this case. In order to determine the time scaling constant 2, one has to do some calculation. It is determined by the asymptotics of the variance of the winding number in an annulus with radii ffl and 1 about 0 and the determination by Kenyon of the corresponding variance for LERW. 33

35 Cororllary (S). Assuming that LERW has a conformally invariant scaling limit, the scaling limit is radial SLE 2. Reminder: this means that the scaling limit path is given by fl(t) = g 1 t ( (t)), where (t) = B(2t), B(t) is BM starting from a random-uniform point, and g t is the solution of g t(z) = g t (z) (t) + g t(z) (t) g t (z) ; g 0(z) = z: 34

36 Percolation is similar Note that the analogue to the combinatorial property for LERW holds for percolation. The fact that the path joins boundary points means that chordal rather than radial SLE is appropriate. The fact that» = 6 for percolation follows by computing Cardy's formula (for a square, the crossing probability is 1=2, by duality). 35

37 The UST Peano and SLE Assuming the conformal invariance of the LERW scaling limit, it follows that the UST Peano path is also conformally invariant, and that a variant of it (to make it start and end in distinct boundary points) is the trace of chordal SLE 8. 36

38 Simple paths Conjecture. Consider the uniform measure on simple grid paths from 0 to the boundary of U. The scaling limit exists and equal to radial SLE 8. Similarly, the scaling limit of uniform measure on simple grid paths joining two specified boundary points of U is equal to chordal SLE 8 (mapped conformally from H onto U ). 37

39 » = 4 We have seen that SLE 6 describes the scaling limit of critical percolation boundary paths, and that conjecturally, SLE 2 and SLE 8 are scaling limits of the LERW and the UST Peano paths. A process conjectured to converge to SLE 4 is Kenyon's dominodifference contour: There are also candidates for various other». 38

40 Phases of SLE Theorem (Rohde-Schramm). 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 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 never crosses itself. The Hausdorff dimension of the SLE path is conjectured to be 1 +»=8 when» 6 8. We have a proof that the expected number of balls of radius 39

41 ffl needed to cover the trace (within a bounded set) grows like ffl (1+»=8). 40

42 It^o's formula Want to differentiate functions of BM with respect to t: d dt F (B t) =? Even more generally, we may have some process Y t such that dy t = a(t) dt + b(t) db t : About all you need to know about stochastic calculus is It^o's formula: df (Y t ) = F 0 (Y t ) dy t + (1=2) F 00 (Y t ) b(t) 2 dt: 41

43 Calculating with SLE Many things can be calculated about the models using the SLE representation. Theorem (S). Assuming that critical percolation has a conformally invariant scaling limit, the probability that there is a percolation cluster in U that intersects a given arc A of length and separates 0 from the complement of A has limit 1 p (2=3) 1 2F ; 2 ; 3 ; 1 2 ß (1=6) cot2 cot : 2 2 A 0 42

44 Proof (sketch) We consider chordal SLE» joining the endpoints of A. We map conformally to the upper half plane by a map which takes the endpoints of A to 0 and 1. Recall the maps g t given by the t g t = 2 g t t : For every point z, there is a first time fi z when g t (z) hits the singularity t. This is also the first time t when the SLE path fl(t) = g 1 t ( t ) separates z from 1. 43

45 Proof (cont.) z We are interested in the probability of the event Q z that the loop around z which fl closes at time fi z is positively oriented around z. That event is equivalent to Re(g t (z) t ) lim t%fi z Im(g t (z) t ) = +1; and the limit is 1 if the loop is negatively oriented. Let h(z) = P[Q z ]. 44

46 Proof (cont.) By the Markov property for BM h(g t (z) t ) is a Martingale. Consequently, It^o's formula gives» xh + t g t (z) = 0 : By the scaling property, h(z) depends only on the direction z=jzj. Since h is a function of one real variable, the above PDE reduces to an ODE. (We set t = 0.) The ODE can be solved for h. 45

47 SLE» Summary» conj process dim magic 0 line seg 1 2 LERW 5=4 Wilson's alg 4 domino difference 6=4 critical 6 percolation boundary 7=4 locality 8 UST Peano path 8=4 space filling Other values of» 2 [0; 8] probably correspond to boundaries of critical random cluster measures. 46

48 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 and Wendelin Werner. 47

49 Brownian intersection exponents Consider BM in the plane. The simplest BM exponent is ο(1; 1): R P[B B 0 = ;] = R ο(1;1)+o(1) : 48

50 Significance of the exponents The exponents encode much information about BM and SRW. For example, the probability that two SRW paths of n steps each starting from zero will not intersect again decays like n ο(1;1)=2 (Burdzy-Lawler). The dimension of the set of cut points of B[0; 1] is a.s. 2 ο(1; 1) (Lawler). 49

51 Determination of the exponents The values of the exponents ο(1; 1;:::;1) have been conjectured by Duplantier-Kwon. We prove a generalization of this: Theorem (Lawler-S-Werner). ο(n 1 ;n 2 ;:::;n k ) = p ( 24n p :::+ 24n k + 1 k) 2 4 : 48 Corollary. The Hausdorff dimension of the set of cut points of B[0; 1] is a.s. 3=4. 50

52 Brownian frontier Theorem (LSW). The Hausdorff dimension of the outer boundary of B[0; 1] is a.s. 4=3. (As conjectured by Mandelbrot.) 51

53 Brownian frontier exponent It was an earlier result of Lawler that the dimension of the frontier is equal to 2 ο(2; 0), where ο(2; 0) is defined as exponent of decay for the probability that two independent BM's starting at 1 will not separate 0 from 1 before hitting the circle of radius R. The event defining ο(2; 0). 52

54 BM exponents and SLE 6 Lawler and Werner had an earlier paper showing that the BM exponents are the same as for other processes satisfying certain axioms. Our proof generally followed that strategy, and the main steps were to show that the (slightly modified) axioms are satisfied and to calculate the exponents for SLE. Now, we have a better understanding of the relation between BM and SLE 6. 53

55 BM frontier and SLE 6 For domains other than H or U, define SLE by mapping conformally. Then SLE is (trivially) conformally invariant. SLE 6 is also local. This means that up to time change, the SLE 6 trace does not feel where the boundary of the domain is, except when touching it. Locality easily follows from the convergence of percolation to SLE 6, however we had to work hard to prove locality, because Smirnov's theorem was not established at that time. Conformal invariance + Locality ) BM frontier 54

56 Consider radial SLE 6 from a small circle ffl@u to 1. Let fl be the limit of the trace as ffl! 0. Given a bounded domain D ρ R 2, consider the hitting measure for fl, that is, the probability measure which is the law of the first point of fl Conformal invariance and locality imply that the hitting measure for fl is the same as for BM starting from 0. Stop fl and BM when we hit the unit Let Y fl be the set of points separated by fl and similarly Y B for the BM. Claim. Y fl and Y B have the same distribution. 55

57 Proof of claim Consider a connected set K ρ U such that 6= ;. The probability that Y fl K 6= ; is the harmonic measure of K as a subset [ K, and the same for Y B. Hence, for every such K, P[Y fl K = ;] = P[Y B K = ;]: This suffices. 56

58 Computing exponents for SLE Do the simplest example of ο(1; 1). In radial SLE, the ODE t g t (z) = g t (z) (t) + g t(z) (t) g t (z) : The time parameter t satisfies g 0 t (0) = exp(t). The distance from 0 to fl[0;t] is about exp( t) (with at most an error by a factor of 4). If we want to measure the probability that another BM (or SLE 6 ) from 0 will without intersecting the current SLE, then what we want is the harmonic measure from 0 in the domain U n fl[0;t]. This is the same as Z length(g t (@U )) = jg 0 (z)jjdzj : 57

59 Conditioned on the SLE, the probability that another BM will not intersect it is jg 0 t (z)jjdzj : So the unconditioned probability is the expectation of this quantity, which can be proved to be approximately Ejg 0 t (1)j ; t = log R: 58

60 Estimating g 0 t(1) Let F ( ; t) := E[ jg 0 t (1)j j log (0) = ] : If we are given in the range [0;s], then the conditioned expected value of Ejg 0 t (1)j is jg 0 s (1)j F (log(g s(1)= (s));t s) : This is by the chain rule and the Markov property. In probabilistic jargon, this means that the above expression is a martingale. If we differentiate with respect to s at s = 0, there cannot be a drift term. It^o's formula then gives a parabolic PDE for F as a function of two variables. The slowest decaying solution for the PDE (as a function of t); that is, the heighest eigenfunction, can be guessed. It is just exp( νt) sin( =2) ; 59

61 where ν(») = 4 +» + p (» 4) : When we take» = 6, we obtain the exponent ο(1; 1) = ν(6) = 5=4. 60

62 What next? Open problem: 1. Prove that the LERW is conformally invariant. (Also gives UST and UST Peano). 2. Prove Smirnov's theorem for other percolation models. 3. There's a conjectured duality for SLE, where if»» 0 = 16 and» > 4 then SLE» 0 describes" the outer boundary of SLE». 4. SLE 8=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 discrete setting. 61