Some percolation processes with infinite range dependencies

Transcription

1 Some percolation processes with infinite range dependencies Marcelo Hilário UFMG (Belo Horizonte) and Unige (Genève) partialy joint with H. Duminil-Copin (Genève), G. Kozma (Tel-Aviv), V. Sidoravicius (Rio de Janeiro) and A.Teixeira (Rio de Janeiro) 2015 Bristol

2 Bernoulli Percolation Z d -lattice = (V (Z d ), E(Z d )). p [0, 1]. Declare each site (or edge) independently { open with prob. p closed with prob. 1 p. P p corresponding law in {0, 1} Zd (or {0, 1} E(Zd) ). Important events {0 B(n)} {0 } C RL (τn, n) C T B (τn, n).

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4 The phase transition I p 7 Pp (0 ) is a non-decreasing function. I Critical point: pc (Zd ) = inf{p [0, 1]; Pp (0 )} > 0. I Phase transition: For d 2, 0 < pc (Zd ) < 1. only finite components 0 subcritical phase unique infinite component pc (Zd ) supercritical phase 1

5 Exponential decay of connectivity How does the quantity {0 B(n)} behaves as a function of n? Subcritical phase: If p < p c (Z d ), then there exists α = α(p, d) > 0 such that P p ({0 B(n)) e α(p)n. (Menshikov 86, Aizenman & Barsky 87). Supercritical phase: If p > p c (Z d ), then there exists σ = σ(p, d) > 0 such that P p ({0 B(n) ) e σ(p)n. (Chayes, Chayes & Newman 87). only infinite components unique infinite component 0 exponential decay p c (Z d exponential decay ) 1

6 Drilling a wooden cube or playing with the Oskar s puzzle

7 Coordinate Percolation Z d -lattice, d 3. {e 1,..., e d } standard orthonormal basis. p 1,..., p d [0, 1] intensity parameters. Remove at random lines parallel to e i with probability p i independently. ne 1 ne 3 L = set of removed sites. V = Z d \L vacant set. p = (p 1,..., p d ). P p = law of the process. ne 2

8 Phase transition ne 1 S 2 (n 0 ) n P 2 P 3 k P 1 n 0 0 n 0 C 1 (0) 0 B d 1 1 (n 0 ) ne 3 P 1 ne 2 Existence of the subcritical phase Existence of the supercritical phase

9 Phase transition Theorem (H., Sidoravicius, 11) Assume that p i < p c (Z d 1 ) for some i {1,..., d}, and that p j 1 for some j {1,..., d}\{i} then P p ({0 }) = 0. (1) On the other hand if p 1,..., p d are sufficiently close to 1, then P p ({0 }) > 0. (2) Theorem (H., Sidoravicius, 11) Let N be the number of infinite connected components. Almost surely under P p, N is a constant random variable taking values in the set {0, 1, }.

10 Decay of correlations Theorem (H., Sidoravicius 11) If p i < p c (Z d 1 ) and p j < p c (Z d 1 ) for some i j {1,..., d}, then there exists a constant ψ = ψ(p, d) > 0 such that, for n 0, P p ({0 B(n)}) e ψ(p,d)n. (3) Theorem (H., Sidoravicius 11) Let d = 3. Assume that p 2 > p c (Z 2 ), p 3 > p c (Z 2 ) and 0 < p 1 < 1. Then, there exists constants α(p) > 0 and α (p) > 0 such that, for all n 0, P p ({0 B(n), 0 }) α (p)n α(p). (4)

11 Proof of the power-law decay

12 More about the phase transition for d = 3. d = 3, p c (Z 2 ) = critical point for site oriented percolation in Z 2. Theorem (H., Sidoravicius 11) If p i > p c (Z 2 ) 1/3 for all i = 1, 2, 3, then P p ({0 }) > 0. If p 2 > p c (Z 2 ) and p 3 > p c (Z 2 ), then there exists ɛ = ɛ(p 2, p 3 ) > 0 such that for all p 1 > 1 ɛ, P p ({0 }) > 0. p 1 = p 2 = p 3 = p. Define p = critical point. Open question: show that p > p c (Z 2 ). 0 p c (Z 2 ) no infinite components? p infinite components 1 p

13 Π What happens when p 1 = p 2 = p 3 = p c (Z 2 ) + δ with δ 0? p Simulation by K.J. Schrenk, N.A.M Araújo, H. J. Herrmann Probability of having a crossing from bottom to top in a box with indicated side length. The data suggest that p = (5). Compare: p c (Z 2 ) and p c (Z 2 ) 1/ exponential decay p c (Z 2 ) no infinite components polynomial decay p infinite components 1 p

14 Cilinders Percolation R d, d 3, L = space of lines of R d. Construct a Poisson point process in L Parametrization: l = line parallel to the canonical vector e d. τ x (y) = y + x. (x, θ) R d 1 SO d. Measure: α : R d 1 SO d L (x, θ) θ(τ x (l)). λ: Lebesgue on R d 1 ; ν: Haar on SO d. µ(a) = (λ ν)(α 1 (A)).

15 Cilinders Percolation Ω = set of locally finite point measures on L. u > 0 parameter. P u = law of a PPP in L with intensity u µ. l L C(l) = cylinder of radius one and axis l. z For ω Ω Set of cylinders: L(ω) = C(l). l supp(ω) Vacant set: x y V(ω) = R d \L(ω).

16 Critical point Main goal: To study the connectivity properties of V under P u. u small few cylinders drilled; u large many cylinders drilled. P u [V has an unbounded component] is non increasing in u. u = inf{u > 0; P u [V has an unbounded component ] = 0} Question: 0 < u <? infinite components no infinite components 0 u R

17 Phase transition Theorem (Tykesson, Windisch 11) For d 3, u < ; For d 4, u > 0. d 4, u small V R 2 has an unbounded component. Why to look at V R 2? Duality If the component of V R 2 containing 0 is bounded, then there exists a circuit in L R 2 surrounding the origin. Multi-scale analysis for ruling out the existence of long circuits

18 The three dimensional case Slow decay of correlations: cov(1 x V, 1 y V ) Theorem (Tykesson, Windisch 11) 1 x y d 1 d = 3 is slower!. d = 3, for all u > 0, V R 2 has only bounded connected components P u a.s.. Infinitely many triangles surrounding the origin in L R 2. u small. Look for unbounded connected components beyond V R 2. Avoiding being trapped by few cylinders. Still being use the duality principle.

19 The three dimensional case Idea : Replace V R 2 by V H. H = hexagonal lattice in R 2 with mesh size H = graph of the application x dist(x, H). Theorem (H., Sidoravicius, Teixeira 12) For d = 3, for all u > 0 small enough P u [V H has an unbounded component ] = 1. Show that there are typically no long paths from 0 in L H.

20 The multiscale analysis x 2 γ = 7/6 fixed. a 0 large. x 1 a n = a γ n 1 = aγn 0 (super exponential growth of scales). p n (u) = sup x R 2 E u [A n (x)]. A n (x) = 1{S(x, a n /10) S(x, a n ) in π(l H)}. Show that p n (u) decays very fast with n.

21 The multiscale analysis x 2 x 1 cover intermediate ( spheres with at most c an balls of radius a n 1 10 a n 1 ) p n (u) = sup x R 2 E u [A n (x)] c ( a n a n 1 ) 2 sup Eu [A n 1 (x 1 )A n 1 (x 2 )] suppremum over x 1 and x 2, centre of balls in the coverings.

22 The multiscale analysis p n (u) c ( an a n 1 ) 2 sup Eu [A n 1 (x 1 )A n 1 (x 2 )] c ( an a n 1 ) 2 [pn 1 (u) 2 + error]. Forget about the error: p n (u) c ( an a n 1 ) 2 pn 1 (u) 2. Recursion: p n 1 (u) a 5/2(1 γ) n 1 p n (u) a 5/2(1 γ) n. ( ) ( ) 2 6 ( ) 2 p n (u) c an a [pn 1 n 1 (u) 2 an 1 an q a n a n 1(u) 2 ]. n }{{} q n (u) = sup sup x R 2 l 1,l 2 L error ( an E u [A n (x, ω+δ l1 +δ l2 )] R, p n 1, q n 1 a n 1 ).

23 The multiscale analysis Recursion: a 0 big and u small { p n 1 a 5/2(1 γ) n 1 q n 1 a 3/2(1 γ) n 1 { p n a 5/2(1 γ) n q n a 3/2(1 γ) n Triggering: As u 0 both p 0 (u) and q 0 (u) vanish. The rough shape of H plays a crucial hole for showing that q 0 (u) vanishes (would be false for R 2 ). P u { there exists a circuit in π(l H) surrounding the origin of R 2 P u { the origin belongs to an unbounded component of π(l H) } < 1, } > 0.

24 Brochette percolation Bernoulli edge percolation in Z 2. Choose a random set of vertical lines. Increase the parameter in this set. How does it affect the critical point? Λ Z, deterministic set. E vert (Λ Z) = set of brochettes. p, q [0, 1] parameters. P Λ p,q : open edge e with prob. = { p, if e E vert (Λ Z), q, otherwise.

25 Brochette percolation Make the set of the brochettes random. ξ = {ξ z } z Z i.i.d. Bernoulli(ρ). Λ(ξ) = {j Z : ξ j = 1}. ν(ρ) = law of ξ. P ρ p,q( ) := P Λ(ξ) p,q ( )dν ρ (ξ). Theorem (Duminil-Copin, H., Kozma, Sidoravicius 15) For every ε > 0 and ρ > 0 there exists δ > 0 such that P ρ p c+ε,p c δ (0 ) > 0. Remark: For the rest of the talk, we fix ε and ρ.

26 Enhancements induced by K-syndetic sets Λ Z is k-syndetic if all its gaps have diameter smaller than k. The Aizenman-Grimmett argument (1991) implies that: Proposition If Λ is k-syndetic then for every ε > 0 there exists δ > 0 such that, P Λ p c+ε,p c δ (0 ) > 0. Russo s Formula: For A an increasing event depending on the state of finitely many edges only (e.g.: {0 B(n)}), d dp P p(a) = e P p (e is pivotal for A), where, {e is pivotal for A} = {ω e A, ω e / A}.

27 The Aizenman-Grimmett argument By Russo s Formula we have: p PΛ p,q(0 B(n)) = q PΛ p,q(0 B(n)) = f E vert(λ Z) e/ E vert(λ Z) P Λ p,q(f is piv. for 0 B(n)). P Λ p,q(e is piv. for 0 B(n)). By local modification arguments, using that Λ is k-syndetic: P Λ p,q(f(e) piv. for 0 B(n)) c(k, p, q)p Λ p,q(e piv. for 0 B(n)). This ultimately leads to: q PΛ p,q(0 B(n)) c(k, p, q) p PΛ p,q(0 B(n)), with c(k, p, q) bounded in a neighbourhood of (p c, p c ).

28 The KSV Theorem Z 2 Edges oriented in the NE and NW sense. Declare columns good independently with probability ρ. Parameter in good lines: p G. Parameter in bad lines: p B. Pρ p G,p B = law Theorem (Kesten, Sidoravicius, Vares, 12) For all p B > 0 and p G > p c (Z 2 ) there exists ρ > 0 such that P ρ p G,p B (oriented infinite path in Z 2 ) > 0.

29 The renormalisation scheme i6 i1 i2 i5 i3 i4 vn(0) Scale n Blocks: v n (z) = [ n, n] 2 +2nz. Lattice: Z 2 n = {v n (z); z even}. c 7 c 6 c 5 c 4 c 3 c 2 c 1 c0 c1 c2 c3 c4 c5 c6 c7 15n 13n 11n 9n 7n 5n 3n n 0 n 3n 5n 7n 9n 11n 13n 15n Columns: c n (i) = {v n (i, j); i + j is even}. c n (i) is good if Λ(ξ) [2n(i 1), 2n(i + 1)] is 2 ρ log 2n-syndetic. v n (z) is good if crossed as above.

30 Crossing probabilities in k-syndetic boxes Lemma lim n P ρ p,q(c n (i) is a good column) = 1. Proposition There exists c > 0 and α > 0 such that for all Λ k-syndetic, P Λ p c+ε,p c (C RL (τn, n)) P pc+ck α(c RL(τn, n)). Lemma lim n P pc+[ 2c ρ log(2n)] α (C RL (τn, n)) = 1. Conclusion: For n large, process in good columns dominates a Bernoulli site percolation. Also one can show that the process in bad columns dominates a Bernoulli site percolation.

31 Proof of the theorem Define p B = and p G = By KSV, there exists ρ ρ such that P p,q(0 ) > 0. Fix n large enough so that: The process of good lines dominates a 1-d i.i.d. Bernoulli(ρ ) sequence. The process of occupied blocks in good lines dominates an Bernoulli percolation. With n fixed, find δ small enough so that, under P ρ p c+ε,p c δ, The process of occupied sites in bad columns still dominates an independent Bernoulli percolation with parameter The process of occupied sites in good columns still dominates an independent Bernoulli percolation with parameter The result follows from KSV.