Finite Connections for Supercritical Bernoulli Bond Percolation in 2D

Transcription

1 Finite Connections for Supercritical Bernoulli Bond Percolation in 2D M. Campanino 1 D. Ioffe 2 O. Louidor 3 1 Università di Bologna (Italy) 2 Technion (Israel) 3 Courant (New York University) Courant Probability Seminar, 11/6/2009

2 Outline Introduction Percolation on Z d Logarithmic Asymptotics of Connectivities Sharp Asymptotics of Connectivities Sketch of Proof Setup Geometry of Finite Connections The Structure of a Cluster Asymptotics for No Intersection of Two Decorated RWs Summary

3 Outline Introduction Percolation on Z d Logarithmic Asymptotics of Connectivities Sharp Asymptotics of Connectivities Sketch of Proof Setup Geometry of Finite Connections The Structure of a Cluster Asymptotics for No Intersection of Two Decorated RWs Summary

4 Percolation Take G = (Z d, E d ) - the integer lattice with nearest neighbor edges. Open each edge with probability p [0, 1] independently. Let B p be the underlying measure. Theorem (Broadbent, Hammersley 1957) For all d > 2, there exists p c (d) (0, 1) such that: { 0 if p < pc (d) B p (0 ) = θ(p) > 0 if p > p c (d)

5 Basic Picture Sub-critical density p < p c (d): All clusters (connected components) are finite. Radii of clusters have exponentially decaying distributions: ξ p (0, ) : B p (0 B(R)) e ξpr. Russo-Menshikov (86), Barskey-Aizenman (87). Super-critical density p > p c (d): Unique infinite cluster B p -a.s. Radii of finite clusters have exponentially decaying distributions: ζ p (0, ) : B p ( 0 B(R)) e ζpr. Chayes 2 -Newman (87), C 2 -Grimmett-Kesten-Schonmann (89).

6 Connectivities The point-to-point connectivity function is defined as: τ p (x, y) = B p (x y) ; x, y Z d.

7 Connectivities The point-to-point connectivity function is defined as: τ p (x, y) = B p (x y) ; x, y Z d. Sub-critical case: Subadditivity (via FKG of B p ) and exponential decay of cluster radius distribution imply: Theorem Assume p < p c (d). Then, for all x R d : ξ p (x) = lim n 1 n log τ p(0, nx ) is well-defined, convex and homogeneous function that is strictly positive on R d \ {0}. In other words ξ p is a norm on R d, called the inverse correlation norm.

8 Connectivities - cont d Super-critical case: FKG gives a uniform positive lower bound for all x, y Z d : B p (x y) B p (x )B p (y ) = θ 2 (p) > 0.

9 Connectivities - cont d Super-critical case: FKG gives a uniform positive lower bound for all x, y Z d : B p (x y) B p (x )B p (y ) = θ 2 (p) > 0. Therefore, define the finite (truncated) connectivity function: f τp f (x, y) = B p(x y) = B p ( x y) ; x, y Z d.

10 Connectivities - cont d Super-critical case: FKG gives a uniform positive lower bound for all x, y Z d : B p (x y) B p (x )B p (y ) = θ 2 (p) > 0. Therefore, define the finite (truncated) connectivity function: f τp f (x, y) = B p(x y) = B p ( x y) ; x, y Z d. Theorem Assume p / {0, p c (d), 1}. Then, for all x R d : ζ p (x) = lim n 1 n log τ f p(0, nx ) is well-defined, homogeneous and strictly positive on R d \ {0}. This is the finite (truncated) inverse correlation function.

11 In other words: Logarithmic Scale Asymptotics B p (x y) e ξp(θ) y x 2 and B p (x f y) e ζp(θ) y x 2 for all x, y Z d as y x, where θ = (x y)/ x y 2.

12 In other words: Logarithmic Scale Asymptotics B p (x y) e ξp(θ) y x 2 and B p (x f y) e ζp(θ) y x 2 for all x, y Z d as y x, where θ = (x y)/ x y 2. Some relations: ξ p = ξ p (e 1 ). ζ p = ζ p (e 1 ). If d = 2, p > p c (2) = 1 2 then ζ p = 2ξ 1 p. Chayes-Chayes-Grimmett-Kesten-Schonmann (89).

13 In other words: Logarithmic Scale Asymptotics B p (x y) e ξp(θ) y x 2 and B p (x f y) e ζp(θ) y x 2 for all x, y Z d as y x, where θ = (x y)/ x y 2. Some relations: ξ p = ξ p (e 1 ). ζ p = ζ p (e 1 ). If d = 2, p > p c (2) = 1 2 then ζ p = 2ξ 1 p. Chayes-Chayes-Grimmett-Kesten-Schonmann (89). Want sharp asymptotics: B p (x y)? and B p (x f y)?

14 Sharp Asymptotics - Subcritical Case For all d 2, p < p c (d), x, y Z d : B p (x y) A p (θ) y x (d 1)/2 2 e ξp(θ) y x 2 as y x. Campanino-Chayes-Chayes (88) (y x is on the axes). Campanino-Ioffe (02) (all y x). With the Gaussian correction this is called Ornstein-Zernike Behavior. after the work of L.Ornstein and F.Zernike.

15 Sharp Asymptotics - Supercritical Case d 3: For all p > p c (d), x, y Z d, it is expected: B p (x f y) Ãp(θ) y x (d 1)/2 2 e ζp(θ) y x 2 as y x. Verified for p p c (d) and y x on axes. Braga-Procacci-Sanchis (04).

16 Sharp Asymptotics - Supercritical Case d 3: For all p > p c (d), x, y Z d, it is expected: B p (x f y) Ãp(θ) y x (d 1)/2 2 e ζp(θ) y x 2 as y x. Verified for p p c (d) and y x on axes. Braga-Procacci-Sanchis (04). d = 2:

17 Sharp Asymptotics - Supercritical Case d 3: For all p > p c (d), x, y Z d, it is expected: B p (x f y) Ãp(θ) y x (d 1)/2 2 e ζp(θ) y x 2 as y x. Verified for p p c (d) and y x on axes. Braga-Procacci-Sanchis (04). d = 2:?

18 A Related Model - Nearest-Neighbor Ising

19 A Related Model - Nearest-Neighbor Ising Exactly solvable in d = 2 (Onsager (44)). Explicit formulas for (truncated) k-point correlation functions at all temperatures (Wu, McCoy, Tracy, Potts, Ward, Montroll ( 70)). Theorem (Cheng-Wu, Wu) If β < β c (2) then σ x ; σ y β σ x σ y β A β (θ) y x 1/2 2 e ξ β(θ) y x 2 and if β > β c (2) then: σ x ; σ y T β σ xσ y β σ x β σ y β Ãβ(θ) y x 2 2 e ζ β(θ) y x 2 for all x, y Z 2 as y x. No OZ Behavior in d = 2 below the critical temperature!

20 Sharp Asymptotics - Supercritical Case d 3: For all p > p c (d), x, y Z d, it is expected: B p (x f y) Ãp(θ) y x (d 1)/2 2 e ζp(θ) y x 2 as y x. Verified for p p c (d) and y x on axes. Braga-Procacci-Sanchis (04). d = 2:?

21 Sharp Asymptotics - Supercritical Case d 3: For all p > p c (d), x, y Z d, it is expected: B p (x f y) Ãp(θ) y x (d 1)/2 2 e ζp(θ) y x 2 as y x. Verified for p p c (d) and y x on axes. Braga-Procacci-Sanchis (04). d = 2: Theorem (Campanino, Ioffe, L. (09)) For all p > p c (2) = 1/2, x, y Z 2 : B p (x f y) Ãp(θ) y x 2 2 e ζp(θ) y x 2 as y x.

22 Outline Introduction Percolation on Z d Logarithmic Asymptotics of Connectivities Sharp Asymptotics of Connectivities Sketch of Proof Setup Geometry of Finite Connections The Structure of a Cluster Asymptotics for No Intersection of Two Decorated RWs Summary

23 Setup Dual lattice. In d = 2 there is an isomorphic dual Z 2. Set: b is open b is close. The dual model is Percolation with p = 1 p. We assume p < p c (2) and find B p (x f y ). However, we ll express this event mainly using direct bonds. For simplicity: x = 0, y = 0 + (N, 0) N.

24 Setup Dual lattice. In d = 2 there is an isomorphic dual Z 2. Set: b is open b is close. The dual model is Percolation with p = 1 p. We assume p < p c (2) and find B p (x f y ). However, we ll express this event mainly using direct bonds. For simplicity: x = 0, y = 0 + (N, 0) N. A bit of notation. Cl m,r (x, y) - The (possibly empty) cluster that contains x, y and uses only edges in the strip [m, r] Z.

25 Geometry of Finite Connections { } 0 f x N = {0 x N } { direct loop C N around 0 and x N }. C N γ 0 N

26 Decomposition of Finite Connection Event Idea: Find a geometric decomposition which is both unique and the sets of bonds that define each pieces are disjoint.

27 Decomposition of Finite Connection Event Idea: Find a geometric decomposition which is both unique and the sets of bonds that define each pieces are disjoint. In our case: Let C N be the inner most loop that contains Cl(0, N ). Cut along the left-most line H m and right-most line H N r that intersect C N exactly twice: C N H m = {x, v} ; C N H N r = {u, y} Get 3 pieces: L([v, x]), A([v, x], [u, y]), R([u, y]).

28 Decomposition - cont d C N γ 0 N ( ) B p 0 f x N

29 Decomposition - cont d C N y x γ 0 N v u B p ( 0 f x N l) = B p (I([v, x], [u, y])) + B p (I( )) x,v,y,u N r

30 Decomposition - cont d y y x x 0 N v v u u ( ) B p 0 f x N = x,v,y,u B p (I([v, x], [u, y])) + B p (I( )) = B p (L([v, x])) B p (A([v, x], [u, y])) B p (R([u, y])) + B p (I( )) x,v,y,u

31 Decomposition - cont d In fact: ( ) B p 0 f x N B p (L([v, x])) B p (A([v, x], [u, y]))b p (R([u, y])) x, v log N N y, N u log N and, modulo the exponential decay, the asymptotics will come from B p (A([v, x], [u, y]))

32 Two Disjoint Boundary Clusters A([v, x], [u, y]) = {..., x y, v u, Cl m,n r (v, u) Cl m,n r (x, y) = } x Cl m,n r (x, y) y v u H m Cl m,n r (v, u) H N r

33 Two Disjoint Boundary Clusters A([v, x], [u, y]) = {..., x y, v u, Cl m,n r (v, u) Cl m,n r (x, y) = } A([v, x], [u, y]) = {..., x y, v u, Cl m,n r (v, u) γ up (Cl m,n r (x, y)) = } x γ up Cl m,n r (x, y) y v u H m Cl m,n r (v, u) H N r

34 Two Disjoint Boundary Clusters A([v, x], [u, y]) = {..., x y, v u, Cl m,n r (v, u) Cl m,n r (x, y) = } A([v, x], [u, y]) = {..., x y, v u, Cl m,n r (v, u) γ up (Cl m,n r (x, y)) = } x γ up Cl m,n r (x, y) y Exploration of Cl m,n r (v, u) and γ up (Cl m,n r (x, y)) uses different bonds. v u H m Cl m,n r (v, u) H N r

35 Two Disjoint Boundary Clusters A([v, x], [u, y]) = {..., x y, v u, Cl m,n r (v, u) Cl m,n r (x, y) = } A([v, x], [u, y]) = {..., x y, v u, Cl m,n r (v, u) γ up (Cl m,n r (x, y)) = } x γ up Cl m,n r (x, y) y Exploration of Cl m,n r (v, u) and γ up (Cl m,n r (x, y)) uses different bonds. = We can sample the clusters independently: B p (A(... )) = B p (A(... )) v H m Cl m,n r (v, u) u H N r

36 The Structure of One Cluster We would like to have some geometric decomposition of a cluster Cl(x, y).

37 The Structure of One Cluster We would like to have some geometric decomposition of a cluster Cl(x, y). Assume x = 0, y = (N, 0).

38 The Structure of One Cluster We would like to have some geometric decomposition of a cluster Cl(x, y). Assume x = 0, y = (N, 0). Definition: H m is an α-cone-cut-line and z H m is an α-cone-cut-point of Cl(x, y) if Cl(x, y) (z C α ) (z + C α ). x z C α z z + C α y where: C α = {x = (t, x) : x αt}. H m

39 The Structure of One Cluster We would like to have some geometric decomposition of a cluster Cl(x, y). Assume x = 0, y = (N, 0). Definition: H m is an α-cone-cut-line and z H m is an α-cone-cut-point of Cl(x, y) if Cl(x, y) (z C α ) (z + C α ). x z C α z z + C α y where: C α = {x = (t, x) : x αt}. H m There is a well-defined irreducible decomposition of Cl(x, y) along cone-cut-lines:

40 Cluster Decomposition - An Illustration x = 0 y = (N, 0)

41 Cluster Decomposition - An Illustration x = 0 y = (N, 0)

42 Cluster Decomposition - An Illustration x = 0 Γ f y = (N, 0) Γ 1 Γ 2 Γ 3 C3 bf Γ b

43 Cluster Decomposition - An Illustration σ f σ b σ σ 3 1 σ 2 S3 bf x = 0 Γ f y = (N, 0) Γ 1 Γ 2 Γ 3 C3 bf Γ b

44 Cluster Decomposition - An Illustration σ f σ b σ σ 3 1 σ 2 S3 bf x = 0 Γ f y = (N, 0) Γ 1 Γ 2 Γ 3 C3 bf Γ b

45 Cluster Decomposition - cont d Let F be the set of all pieces that can appear between any two succeeding cone-cut-points in any decomposition.

46 Cluster Decomposition - cont d Let F be the set of all pieces that can appear between any two succeeding cone-cut-points in any decomposition. A piece Γ F comes with an offset vector σ = σ(γ), which is the difference between its left and right cut-points.

47 Cluster Decomposition - cont d Let F be the set of all pieces that can appear between any two succeeding cone-cut-points in any decomposition. A piece Γ F comes with an offset vector σ = σ(γ), which is the difference between its left and right cut-points. Similarly, define F b and F f for all possible initial and final pieces. F = Γ σ σ b F b = F f = σ f Γ b Γ f

48 Then, Cluster Decomposition - cont d B p (Cl(x, y) ) = B p (no cone-cut-lines) + where the sum is over all: Γ b F b, Γ i F for i = 1,..., n and all n, Γ f F f, such that: y = x + σ b + σ σ n + σ f. Γ B p ({Γ b })B p ({Γ 1 }) B p ({Γ n })B p ({Γ f })

49 Then, Cluster Decomposition - cont d B p (Cl(x, y) ) = B p (no cone-cut-lines) + where the sum is over all: Γ b F b, Γ i F for i = 1,..., n and all n, Γ f F f, such that: y = x + σ b + σ σ n + σ f. In fact,. Γ B p ({Γ b })B p ({Γ 1 }) B p ({Γ n })B p ({Γ f }) B p (Cl(x, y) ) Γ B p ({Γ b })B p ({Γ 1 }) B p ({Γ n })B p ({Γ f })

50 We can grow Cl(x, y) iteratively: 1. Draw Γ b F b w.p. B p ({Γ b }). Draw Γ f F f w.p. B p ({Γ f }). 2. Set C 0 =, S 0 = 0. Growing a cluster 3. At each step m: 3.1 Draw Γ m F w.p. B p({γ m}). 3.2 C m = C m 1 Γ m ; S m = S m 1 + σ m. 3.3 Cm bf = {x} Γ b C m Γ f. Sm bf = x + σ b + S m + σ f.

51 We can grow Cl(x, y) iteratively: 1. Draw Γ b F b w.p. B p ({Γ b }). Draw Γ f F f w.p. B p ({Γ f }). 2. Set C 0 =, S 0 = 0. Growing a cluster 3. At each step m: 3.1 Draw Γ m F w.p. B p({γ m}). 3.2 C m = C m 1 Γ m ; S m = S m 1 + σ m. 3.3 Cm bf = {x} Γ b C m Γ f. Sm bf = x + σ b + S m + σ f. If P x is the measure for this decorated RW, then B p (Cl(x, y), cone-cut-lines) = P x ( n s.t S bf n = y).

52 Growing a Cluster - Illustration σ b σ f x = 0 y = (N, 0) Γ S b 0 bf Γ f C0 bf

53 Growing a Cluster - Illustration σ f σ 1 σ b S1 bf x = 0 Γ Γ f y = (N, 0) 1 C1 bf Γ b

54 Growing a Cluster - Illustration σ σ f 2 σ 1 σ b S2 bf Γ 2 x = 0 Γ f Γ C y = (N, 0) 1 2 bf Γ b

55 Growing a Cluster - Illustration σ 3 σ 2 S3 bf σ 1 σ Γ f b Γ 3 C3 bf Γ 2 x = 0 Γ y = (N, 0) 1 Γ b σ f

56 Growing a Cluster - Illustration σ 3 σ 2 S3 bf σ 1 σ Γ f b Γ 3 C3 bf Γ 2 x = 0 Γ y = (N, 0) 1 Γ b σ f σ b σ f x = 0 Sbf y = (N, 0) 0 Γ b Γ f C bf 0

57 Growing a Cluster - Illustration σ 3 σ 2 S3 bf σ 1 σ Γ f b Γ 3 C3 bf Γ 2 x = 0 Γ y = (N, 0) 1 Γ b σ f σ b σ 1 σ f x = 0 S1 bf y = (N, 0) Γ b Γ 1 Γ f C1 bf

58 Growing a Cluster - Illustration σ 3 σ 2 S3 bf σ 1 σ Γ f b Γ 3 C3 bf Γ 2 x = 0 Γ y = (N, 0) 1 Γ b σ f σ b σ 1 σ 2 σ f x = 0 y = (N, 0) Γ 1 Γ 2 S2 bf Γ b Γ f C2 bf

59 Growing a Cluster - Illustration σ 3 σ 2 S3 bf σ 1 σ Γ f b Γ 3 C3 bf Γ 2 x = 0 Γ y = (N, 0) 1 Γ b σ f σ f σ b σ σ 3 1 σ 2 S3 bf x = 0 y = (N, 0) Γ Γ f 1 Γ 2 Γ 3 C Γ 3 bf b

60 Normalizing Steps Γ are not properly defined: Γ F B p ({Γ}) < 1. i.e. the random walk can die.

61 Normalizing Steps Γ are not properly defined: Γ F B p ({Γ}) < 1. i.e. the random walk can die. Tilt to make proper: P(Γ = γ) = P(Γ b,f = γ) = e K σ(γ),e 1 P(Γ = γ). k 1 e K σ(γ),e 1 P(Γb,f = γ).

62 Normalizing Steps Γ are not properly defined: Γ F B p ({Γ}) < 1. i.e. the random walk can die. Tilt to make proper: P(Γ = γ) = P(Γ b,f = γ) = e K σ(γ),e 1 P(Γ = γ). k 1 e K σ(γ),e 1 P(Γb,f = γ). K = ξ p (e 1 )!!

63 Normalizing Steps Γ are not properly defined: Γ F B p ({Γ}) < 1. i.e. the random walk can die. Tilt to make proper: P(Γ = γ) = P(Γ b,f = γ) = e K σ(γ),e 1 P(Γ = γ). k 1 e K σ(γ),e 1 P(Γb,f = γ). K = ξ p (e 1 )!! Then: B p (Cl(x, y) ) K e ξp(e 1) y x,e 1 P x ( n s.t S bf n = y).

64 Back to the Two Boundary Clusters Thus, B p (A([v, x], [u, y])) Ke 2ξp(e 1) y x,e 1 ( ) P x,v n 1, n 2 s.t Sn bf,1 1 = y, Sn bf,2 2 = u and C bf,1 C bf,2 =, n 1 n 2 where under P x,v two decorated RWs: C bf,1, C bf,1 evolve independently starting from (x, v). The prefactor will come, therefore, from the asymptotics of the probability of the RW event above.

65 Asymptotics of No Intersection Let N = y x, e 1 = u v, e 1. Then Lemma As N, uniformly in x, y, u, v D(N): ( ) P x,v n 1, n 2 s.t Sn bf,1 1 = y, Sn bf,2 2 = u and C bf,1 C bf,2 = n 1 n 2 U(x v)u(y u) 1 N 2. U( ) has polynomial growth.

66 Proof of Intersection Asymptotics 1. One (unbiased) RW staying positive: P 0 (S n = y, S k > 0 k) U(y) n P 0(S n = y) ; y n 1/2 Combinatorial proof by Alili-Doney (97).

67 Proof of Intersection Asymptotics 1. One (unbiased) RW staying positive: P 0 (S n = y, S k > 0 k) U(y) n P 0(S n = y) ; y n 1/2 Combinatorial proof by Alili-Doney (97). 2. Starting from x > 0: P x(s n = y, S k > 0 k) U(x)U(y) P x(s n = y) ; x, y n 1/2 n

68 Proof of Intersection Asymptotics 1. One (unbiased) RW staying positive: P 0 (S n = y, S k > 0 k) U(y) n P 0(S n = y) ; y n 1/2 Combinatorial proof by Alili-Doney (97). 2. Starting from x > 0: P x(s n = y, S k > 0 k) U(x)U(y) P x(s n = y) ; x, y n 1/2 n 3. Two independent RWs not intersecting: P x,v(s 1 n = y, S 2 n = u, S 1 k > S 2 k k) U(x v)u(y u) P x(s n = y)p v(s n = u) ; x, v, y, u n 1/2 n

69 Proof of Intersection Asymptotics - cont d 4 Add random time steps: ) P (0,x),(0,v) ( n 1, n 2 s.t Sn 1 1 = (N, y), Sn 2 2 = (N, u), S 1 S 2 = U(x v)u(y u) N 1 1 N ; x, v, y, u N 1/2 N

70 Proof of Intersection Asymptotics - cont d 4 Add random time steps: ) P (0,x),(0,v) ( n 1, n 2 s.t Sn 1 1 = (N, y), Sn 2 2 = (N, u), S 1 S 2 = 5 By entropic repulsion: U(x v)u(y u) N 1 1 N ; x, v, y, u N 1/2 N ( P (0,x),(0,v) C 1 n 1 C2 n = S 1 2 S 2 =,... ) const

71 Proof of Intersection Asymptotics - cont d 4 Add random time steps: ) P (0,x),(0,v) ( n 1, n 2 s.t Sn 1 1 = (N, y), Sn 2 2 = (N, u), S 1 S 2 = 5 By entropic repulsion: U(x v)u(y u) N 1 1 N ; x, v, y, u N 1/2 N ( P (0,x),(0,v) C 1 n 1 C2 n = S 1 2 S 2 =,... ) const 6 Initial and final piece only change the constant: ( ) P (0,x),(0,v) n 1, n 2 s.t Sn 1,bf 1 = (N, y), Sn 2,bf 2 = (N, u), C 1,bf C 2,bf = n 1 n 2 U(x v)u(y u) N 2 ; x, v, y, u N 1/2

72 Outline Introduction Percolation on Z d Logarithmic Asymptotics of Connectivities Sharp Asymptotics of Connectivities Sketch of Proof Setup Geometry of Finite Connections The Structure of a Cluster Asymptotics for No Intersection of Two Decorated RWs Summary

73 Open Questions Do the same for 2D Random Cluster (FK) model (q 2). Supercritical percolation on Z d for d 3 Show OZ behavior for finite connectivities in all directions and all p > p c (d).

74 Thank You Thank you.