Dependent percolation: some examples and multi-scale tools

Transcription

1 Dependent percolation: some examples and multi-scale tools Maria Eulália Vares UFRJ, Rio de Janeiro, Brasil 8th Purdue International Symposium, June 22

2 I. Motivation Classical Ising model (spins ±) in two dimensions H(σ) = J {x,y} σ x σ y x y = σ {, +} Λ ; Λ Z 2 (finite) J {x,y} J > ferromagnetic interaction; x, y Z 2 µ Λ (σ) = Z Λ exp { βh Λ (σ)} (probability measure) Phase transition at sufficiently low temperature multiple limits of µ Λ as Λ Z 2 if β > β c. β c = 2J log( + 2) Peierls (936); Onsager (944); Yang (952)

3 I. Motivation Simulations by Vincent Beffara

4 I. Motivation Simulations by Vincent Beffara

5 I. Motivation McCoy and Wu (968) investigated the effect of random impurities

6 I. Motivation McCoy and Wu (968) investigated the effect of random impurities

7 I. Motivation McCoy and Wu (968) investigated the effect of random impurities Disordered ferromagnets randomly layered environment

8 I. Motivation/Plan Phase transitions for systems in a (randomly) layered environment Growth processes in random environment Some forms of coordinate percolation. Winkler s compatibility problem.

9 I. Motivation/Plan Phase transitions for systems in a (randomly) layered environment Growth processes in random environment Some forms of coordinate percolation. Winkler s compatibility problem. Common basic tool: multi-scale analysis One learns with simpler hierarchical structures

10 I. Motivation/Plan Phase transitions for systems in a (randomly) layered environment Growth processes in random environment Some forms of coordinate percolation. Winkler s compatibility problem. Common basic tool: multi-scale analysis One learns with simpler hierarchical structures Based on joint results with H. Kesten, B. Lima, V. Sidoravicius

11 II. Oriented percolation in a randomly layered environment On Z 2 + := {(x, y) Z Z + : x + y is even} consider the following oriented (NW, NE) site percolation model:

12 II. Oriented percolation in a randomly layered environment On Z 2 + := {(x, y) Z Z + : x + y is even} consider the following oriented (NW, NE) site percolation model: lines H i := {(x, y) Z 2 + : y = i} are declared bad or good with probabilities δ and δ respectively, independently of each other.

13 II. Oriented percolation in a randomly layered environment On Z 2 + := {(x, y) Z Z + : x + y is even} consider the following oriented (NW, NE) site percolation model lines H i := {(x, y) Z 2 + : y = i} are declared bad or good with probabilities δ and δ respectively, independently of each other. Sites on good lines are open with probability p G and sites on bad lines are open with probability p B, all independently of each other.

14 II. Oriented percolation in a randomly layered environment On Z 2 + := {(x, y) Z Z + : x + y is even} consider the following oriented (NW, NE) site percolation model lines H i := {(x, y) Z 2 + : y = i} are declared bad or good with probabilities δ and δ respectively, independently of each other. Sites on good lines are open with probability p G and sites on bad lines are open with probability p B, all independently of each other. Regime of interest: < p B < p c < p G, < δ <. What can we say about occurrence of percolation? (a.s. in the environment...)

15 II. Oriented percolation in a randomly layered environment On Z 2 + := {(x, y) Z Z + : x + y is even} consider the following oriented (NW, NE) site percolation model lines H i := {(x, y) Z 2 + : y = i} are declared bad or good with probabilities δ and δ respectively, independently of each other. Sites on good lines are open with probability p G and sites on bad lines are open with probability p B, all independently of each other. Regime of interest: < p B < p c < p G, < δ <. What can we say about occurrence of percolation? (a.s. in the environment...) Natural Guess: δ large no percolation δ small percolation

16 II. Oriented percolation in a randomly layered environment On Z 2 + := {(x, y) Z Z + : x + y is even} consider the following oriented (NW, NE) site percolation model lines H i := {(x, y) Z 2 + : y = i} are declared bad or good with probabilities δ and δ respectively, independently of each other. Sites on good lines are open with probability p G and sites on bad lines are open with probability p B, all independently of each other. Regime of interest: < p B < p c < p G, < δ <. What can we say about occurrence of percolation? (a.s. in the environment...) Natural Guess: δ large no percolation (easy) δ small percolation

17 II. Oriented percolation in a randomly layered environment

18 II. Oriented percolation in a randomly layered environment

19 . II. Oriented percolation in a randomly layered environment C + () = {v : open oriented path from to v} Let Θ(p G, p B, δ) = P(C + () is infinite ). Theorem (Kesten, Sidoravicius, V.) p G > p c, p B >, δ > so that Θ(p G, p B, δ) > if δ δ. In fact P(C + () is infinite ξ) > a.s. in ξ (ξ configuration of lines)

20 Basic tool: multi-scale analysis Get started with a very simple situation: hierarchical model, L large (depending on p G, p B ), { k, if L k j but L k+ j, ζ j :=, if L j. Replace each entry ζ j = k by k consecutive bad lines (shift the rest to the right) bad walls of thickness k: k consecutive bad lines; such bad walls at distance of order L k from each other.

21 Basic tool: multi-scale analysis This immediately calls to the consideration of rescaled lattices, all similar to the original one, and adapted to deal with bad lines of thickness k. L cl (c suitable, depending on p G )

22 Basic tool: multi-scale analysis This immediately calls to the consideration of rescaled lattices, all similar to the original one, and adapted to deal with bad lines of thickness k. L cl (c suitable, depending on p G ) p G > p so that P(event in the picture seed) ( p G ) 2

23 Basic tool: multi-scale analysis This immediately calls to the consideration of rescaled lattices, all similar to the original one, and adapted to deal with bad lines of thickness k. Renormalized k-sites S k (i,j), (i, j) Z 2 +.

24 Recursively define the notion of a good k-site S k being passable from a (k )-seed.

25 Recursively define the notion of a good k-site S k being passable from a (k )-seed. It should: (a) guarantee the existence of open oriented paths that cross it from bottom to top (b) produce new (k )-seeds on the top, so that the existence of oriented k-passable paths implies the existence of open oriented paths at scale, and can essentially be compared with independent site percolation.

26 Recursively define the notion of a good k-site S k being passable from a (k )-seed. It should: (a) guarantee the existence of open oriented paths that cross it from bottom to top (b) produce new (k )-seeds on the top, so that the existence of oriented k-passable paths implies the existence of open oriented paths at scale, and can essentially be compared with independent site percolation. Want estimate p k P ( ) S k is s-passable (k ) seed, k, ( )

27 Recursively define the notion of a good k-site S k being passable from a (k )-seed. It should: (a) guarantee the existence of open oriented paths that cross it from bottom to top (b) produce new (k )-seeds on the top, so that the existence of oriented k-passable paths implies the existence of open oriented paths at scale, and can essentially be compared with independent site percolation. Want estimate p k P ( ) S k is s-passable (k ) seed, k, ( ) Proposition. Given p B > and p G > p, there exists L large enough such that (*) holds with p k = q k and and where q = p G. q k q 2 k for all k,

28 The above estimate clearly implies that for L large enough P (there exists an infinite cluster starting from the origin) + k= p 3 k >. The main difficulty in pushing the estimate at each step is when one faces the bad wall of larger mass. Planarity plays important role in these arguments. Enlarging the seeds and taking some extra care replace p by p c.

29 seed S k (i, j) D r D r bad layer of mass k D r K Kernel of S k k- S k S (i+, j-) S k (i-, j-)

30 S k- (i, j+) bad line of mass k-2 reverse partition k- S (i, j+) bad line of mass k- k- S (i, j) forward partition bad line of mass k-2 reverse partition

31 Dealing with random layers Step : Devise a suitable grouping procedure Step 2: Perform the recursive (much more involved!) estimates

32 Step : Grouping procedure ξ {, } N sampled from the Bernoulli distribution P δ with low density δ. Γ = {i : ξ i = } - correspond to the bad lines (of level ): L 3 integer.

33 Step : Grouping procedure ξ {, } N sampled from the Bernoulli distribution P δ with low density δ. Γ = {i : ξ i = } - correspond to the bad lines (of level ): L 3 integer. Start with L-runs...

34 Step : Grouping procedure ξ {, } N sampled from the Bernoulli distribution P δ with low density δ. Γ = {i : ξ i = } - correspond to the bad lines (of level ): L 3 integer. Start with L-runs...

35 Step : Grouping procedure ξ {, } N sampled from the Bernoulli distribution P δ with low density δ. Γ = {i : ξ i = } - correspond to the bad lines (of level ): L 3 integer. Start with L-runs... Assuming 64δL 2 < the procedure converges: Each x Γ will be re-incorporated finitely many times a.s.; the final partition is well defined.

36 Step : Grouping procedure On a set Ξ(δ) of full measure we can decompose Γ into sets C i, called clusters, to which an N-valued mass m(c i ) is attributed (m(c i ) C i ) in a way that d(c i, C j ) L min{m(c i ),m(c j )}, for all i, j The C i := C,i are obtained by the (limiting) recursive procedure outlined above. Each constructed cluster has a level (the step when it was born!) and a mass. m m 2 3 m d d M = M m + m 2 + m 3 log L d log L d 2 C C 2 C 3

37 Step 2: Multi-scale analysis for fixed realization of good/bad lines Assume 3 L, 64δL 2 <. χ(ξ) := inf{k : d(c, ) M m(c) for all C C with m(c) > k} with χ(ξ) = if the above set is empty or ξ / Ξ(δ) Then: P δ {ξ : χ(ξ) < } = and P δ {ξ : χ(ξ) = } >. We prove P ( ) C + () is infinite) χ(ξ) < >. Conceptually, the structure is similar to that in the simple hierarchical situation, but: rescaled lattices depend also on ξ; main estimate (drilling through the higher mass) within a good k-site S k is much more involved; our estimates require L somehow larger (L 92 suffices); p k exponentially in k.

38 III. Some related results Bramson, Durrett, Schonmann (99)

39 III. Some related results Bramson, Durrett, Schonmann (99)

40 IV. Coordinate percolation. Winkler s compatibility (η, ξ) a pair of sequences in Ξ = {, } N Allowed to: remove ones from η; remove zeros from ξ Can one map both sequences to the same semi-infinite sequence? If YES, say that (η, ξ) is compatible.

41 IV. Coordinate percolation. Winkler s compatibility (η, ξ) a pair of sequences in Ξ = {, } N Allowed to: remove ones from η; remove zeros from ξ Can one map both sequences to the same semi-infinite sequence? If YES, say that (η, ξ) is compatible. Winkler s compatibility question: Does it exist (p, p) (, ) 2 such that P p P p {(η, ξ) Ξ Ξ : (η, ξ) are compatible} >? P. Gács (24). Recent preprints: Basu, Sly (22), Sidoravicius (22).

42 IV. Coordinate percolation. Winkler s compatibility (η, ξ) a pair of sequences in Ξ = {, } N Allowed to: remove ones from η; remove zeros from ξ Can one map both sequences to the same semi-infinite sequence? If YES, say that (η, ξ) is compatible. Winkler s compatibility question: Does it exist (p, p) (, ) 2 such that P p P p {(η, ξ) Ξ Ξ : (η, ξ) are compatible} >? P. Gács (24). Recent preprints: Basu, Sly (22), Sidoravicius (22). Let p (, ). Say that η Ξ is p-compatible if P p {ξ Ξ : (η, ξ) is compatible} >.

43 IV. Coordinate percolation. Winkler s compatibility Theorem (Kesten, Lima, Sidoravicius, V.) For every ɛ > there exist < p ɛ < and a binary sequence η η ɛ Ξ, such that Z η ɛ is a discrete fractal with Hausdorff dimension d H (Z η ) ɛ, and such that for any p < p ɛ. P p {ξ Ξ : (η, ξ) is compatible} > Notation: Z η = {i : η i = } For the proof exploit a representation as coordinate oriented percolation; essential ingredient: the grouping procedure mentioned before. Move from ξ to ψ Z N + : ψ i representing the length of the corresponding run of s. Ψ = { ψ Z N + : ψ i implies ψ i+ = }.

44 IV. Coordinate percolation. Winkler s compatibility Coordinate percolation process. Oriented graph G = (V, E), where V = Z 2 + E = { vertical n.n., northeast diagonals } oriented upwards Given ζ, ψ Ψ, define the site configuration ω ζ,ψ on G: for v = (v, v 2 ) with v, v 2 { if ζv ψ v2, ω ζ,ψ (v) = otherwise. v V open iff ω ζ,ψ (v) = (ω ζ,ψ (, ) =, ω ζ,ψ (v) = if v v 2 = )

45 Simulations of coordinate percolation (by Lionel Levine)

46 Simulations of coordinate percolation (by Lionel Levine)

47 Simulations of coordinate percolation (by Lionel Levine)

48 Simulations of coordinate percolation (by Lionel Levine)

49 Simulations of coordinate percolation (by Lionel Levine)

50 Simulations of coordinate percolation (by Lionel Levine)

51 IV. Coordinate percolation. Winkler s compatibility Figure : oriented cluster of the origin For the compatibility question, we need more than an open oriented path. A vertex v = (v, v 2 ) with v, v 2 is heavy if ψ v2.

52 IV. Coordinate percolation. Winkler s compatibility Permitted path: does not cross two heavy vertices with the same first coordinate. Figure 2: open permitted path from the origin

53 IV. Coordinate percolation. Winkler s compatibility Lemma Let ζ, ψ Ψ. If there exists an infinite open permitted path π starting from the origin for the percolation configuration ω ζ,ψ, then the pair (ζ, ψ) is compatible.

54 IV. Coordinate percolation. Winkler s compatibility M-spaced sequences M 2 integer. A sequence ψ Ψ is M-spaced if: a) i j (ψ) M j for all j, where i j (ψ) = inf{n N : ψ n j} (+ if { } = ) b) j i M min{ψ i,ψ j }, for all i < j. Ψ M := {ξ Ψ : ξ is M-spaced}

55 IV. Coordinate percolation. Winkler s compatibility M-spaced sequences M 2 integer. A sequence ψ Ψ is M-spaced if: a) i j (ψ) M j for all j, where i j (ψ) = inf{n N : ψ n j} (+ if { } = ) b) j i M min{ψ i,ψ j }, for all i < j. Ψ M := {ξ Ψ : ξ is M-spaced} Theorem Let L 2 and M 3(L + ) be integers, ζ(l) the hierarchical sequence as before, and ψ Ψ M. Then the configuration ω ζ(l),ψ has an infinite open permitted path π starting from the origin. Corollary If ψ Ψ M with M = 3(L + ) the pair (ζ(l), ψ) is compatible.

56 IV. Coordinate percolation. Winkler s compatibility Using this and the grouping lemma discussed before, one gets: Theorem Let L 2 and M = 3(L + ). If p < 64M 2, ζ(l) Ψ is given by ( ζ(l)) j = { 3M k, if L k j and L k+ j,, if L j, and η(l) is the corresponding binary sequence, then P p {ξ Ξ : (η(l), ξ) is compatible} >. Remark. The statement about the zero set of η(l) is simple to verify, by classical results (Barlow and Taylor).

57 V. Related problems Winkler s Clairvoyant Demon problem

58 V. Related problems Winkler s Clairvoyant Demon problem

59 V. Related problems Winkler s Clairvoyant Demon problem

60 V. Related problems Winkler s Clairvoyant Demon problem

61 V. Related problems Winkler s Clairvoyant Demon problem

62 V. Related problems Winkler s Clairvoyant Demon problem b m b b 2 r r 2 n Formulation as oriented percolation problem - Noga Alon r

63 V. Related problems Winkler s Clairvoyant Demon problem b m b b 2 r r 2 n Formulation as oriented percolation problem - Noga Alon r

64 V. Related problems Winkler s Clairvoyant Demon problem Negative answer if n = 2 or n = 3; n 4 - open question Positive answer in the unoriented case for n 4: P Winkler (2); Balister, Bollobas, Stacey (2) For the oriented model: Gács (2): The Clairvoyant Demon has a hard task...

65 V. Related problems - Stretched lattices: Jonasson, Mossel, Peres ; Hoffman - Unoriented percolation; Potts model: Kesten, Lima, Sidoravicius, V. - Percolation of words: Grimmett, Liggett, Richthammer (28), Lima (28, 29) - Rough isometries: Peled (2), recent work: Basu, Sly, Sidoravicius.

66 References On the compatibility of binary sequences. Vares. (preprint on arxiv) H. Kesten, B. Lima, V. Sidoravicius, M.E. Oriented percolation in a random environment. H. Kesten, V. Sidoravicius, M.E. Vares. (preprint on arxiv) Dependent percolation on Z 2. H. Kesten, B. Lima, V. Sidoravicius, M.E. Vares. (preprint) Lipschitz embeddings of random sequences. R. Basu, A. Sly. (preprint on arxiv)