Stochastic domination in space-time for the supercritical contact process

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1 Stochastic domination in space-time for the supercritical contact process Stein Andreas Bethuelsen joint work with Rob van den Berg (CWI and VU Amsterdam) Workshop on Genealogies of Interacting Particle Systems, IMS, Singapore, August 10th 2017 (S.A. Bethuelsen) 1 / 15

2 The contact process Let G = (V, E) be a connected graph with bounded degree. Let λ (0, ). The contact process on G with infection parameter λ is the IPS on {0, 1} V with local transition rates given by η η x at rate { 1, if η(x) = 1; λ {y : {x,y} E} η(y), if η(x) = 0, where η x is defined by η x (y) := η(y) for y x, and η x (x) := 1 η(x). (S.A. Bethuelsen) 2 / 15

3 Some well-known properties The state 0 where 0(x) = 0 for all x V is an absorbing state. Started from the state 1, where 1(x) = 1 for all x V, the contact process converges toward a stationary measure called the upper invariant measure. We denote this measure by ν λ. For G infinite: there exists λ c (0, ) such that { νλ = δ 0, if λ < λ c; ν λ is non-trivial if λ > λ c. The contact process is said to be supercritical when λ > λ c. (S.A. Bethuelsen) 3 / 15

4 Stochastic domination Associate to {0, 1} V the partial ordering, where η ξ if η(x) ξ(x) for all x V. An event B is increasing if η B implies that ξ B for all ξ η. A measure µ stochastically dominates ν if µ(b) ν(b) for all B increasing. Equivalently, if there exists a coupling P of µ and ν such that P(X µ Y ν ) = 1. Theorem (Liggett and Steif (2006)) Consider the contact process on Z d, d 1, with λ > λ c. Then there exists ρ (0, 1) such that ν λ stochastically dominates a Bernoulli product measure with density ρ. (S.A. Bethuelsen) 4 / 15

5 Correlation inequalities A measure µ on {0, 1} V is said to be: 1 positively associated if µ(b 1 B 2 ) µ(b 1 )µ(b 2 ) for any two increasing events B 1, B 2. 2 downward FKG if for every finite Λ V, the measure µ( η 0 on Λ) is positively associated. 3 FKG if for every finite Λ V and σ {0, 1} V, the measure µ( η σ on Λ) is positively associated. - Liggett (1994) showed that the contact process is not FKG (at least in some regime of λ for the process on Z). - Van den Berg, Häggström and Kahn (2006) proved that ν λ is downward FKG. (S.A. Bethuelsen) 5 / 15

6 Stochastic domination for dfkg measures Theorem (Liggett and Steif (2006)) Let µ be a translation invariant measure on {0, 1} Z which is dfkg. Then the following are equivalent: 1 µ stochastically dominates a Bernoulli product measure with density ρ. 2 µ(η 0 on {1, 2,..., n}) (1 ρ) n for all n. 3 For all disjoint, finite subsets Λ and of {1, 2, 3,... }, we have µ (η(0) = 1 η 0 on Λ, η 1 on ) ρ. - Liggett and Steif also proved a generalization of this theorem for measures on {0, 1} Zd, d 2. (S.A. Bethuelsen) 6 / 15

7 Stochastic domination in space-time For α [0, ), let (ξ t ) t 0 be the independent spin-flip process on {0, 1} V with local transition rates given by η η x at rate { 1, if η(x) = 1; α, if η(x) = 0. This process is uniquely ergodic with the Bernoulli product measure with density ρ = α 1+α, denoted here by µ ρ, as invariant measure. Main question: Does there, for some α > 0, exists a coupling P of (ξ t ) and (η t ) such that, P (η t (x) ξ t (x) for all (x, t) V [0, )) = 1? (1) If not, does (1) possibly hold on certain subsets of V? (S.A. Bethuelsen) 7 / 15

8 The answer depends on the graph Let d : V V N be the graph distance. 1 We say that the graph G = (V, E) is amenable if e U inf = 0. U V,U finite U 2 We say that G has subexponential growth if, for some o V, lim inf n {x V : d(o, x) n} 1/n = 1. 3 We say that V has positive density if, for some o V, lim inf n {x : d(o, x) n} {y V : d(o, y) n} > 0. (S.A. Bethuelsen) 8 / 15

9 Negative answer #1: amenable graphs Proposition (van den Berg, B. (2017)) Consider the contact process on an amenable graph G with λ > 0. Then there does not exist a coupling P of (ξ t ) and (η t ) such that, unless α = 0. P (η t (x) ξ t (x) for all (x, t) V [0, )) = 1 - Liggett and Steif (2006) proved this for the case V = Z. (S.A. Bethuelsen) 9 / 15

10 Negative answer # 2: graphs of subexponential growth Proposition (van den Berg, B. (2017)) Consider the contact process on a graph of subexponential growth with λ > 0. Let V have positive density. Then there does not exist a coupling P of (ξ t ) and (η t ) such that, unless α = 0. P (η t (x) ξ t (x) for all (x, t) [0, )) = 1 - note that there are amenable graphs with exponential growth and which are not covered by the above statement. (S.A. Bethuelsen) 10 / 15

11 Positive answer # 1: domination at a single site For x V, denote by τ x := inf{t 0: η x t 0}. Theorem (van den Berg, B. (2017)) Consider the contact process on a connected graph G = (V, E) having bounded degree with λ > 0. Let x V for which P (τ x = ) > 0 and such that, for some C, c > 0, P (s < τ x < ) Ce cs, for all s 0. Then there exists α = α(λ) > 0 and a coupling P of (η t ) and (ξ t ) initialised from ν λ and µ α/(1+α) respectively, such that P (η t (x) ξ t (x) for all (x, t) {o} [0, )) = 1, (S.A. Bethuelsen) 11 / 15

12 Positive answer # 2: domination on trees Theorem (van den Berg, B. (2017)) Consider the contact process on T d, d 2, and λ > λ c (Z). Then there exists V of positive density, α > 0, and a coupling P of (ξ t ) and (η t ) initialised from ν λ and µ α/(1+α) respectively such that P (η t (x) ξ t (x) for all (x, t) [0, )) = 1. (2) - the theorem also holds for many other non-amenable graphs. - Question: Does the theorem hold on the entire graph T d? (S.A. Bethuelsen) 12 / 15

13 Positive answer # 3: domination on finite sets Theorem (van den Berg, B. (2017)) Consider the contact process on a connected graph G = (V, E) having bounded degree with λ > 0. Let V finite for which, for all x, P (τ x = ) > 0 and such that, for some C, c > 0, P (s < τ x < ) Ce cs, for all s 0. Then there exists α = α(λ) > 0 and a coupling P of (η t ) and (ξ t ) initialised from ν λ and µ α/(1+α) respectively, such that P (η t (x) ξ t (x) for all (x, t) N) = 1, (S.A. Bethuelsen) 13 / 15

14 An application: cone-mixing on slabs Theorem (van den Berg, B. (2017)) The contact process on Z d, d 1, with λ > λ c, projected onto Z d 1 {0} Z, is cone-mixing. That is, for all θ (0, 1 2 π), lim t sup A F <0,B F θ t P(A)>0 P(B A) P(B) = 0. where F <0 is the σ-algebra generated by the (discrete-time) lower half-space and Ft θ is the σ-algebra generated by the (discrete-time) forward cone with declination proportional to θ. - Cone mixing is an important property in the study of random walks in (dynamic) random environment. (S.A. Bethuelsen) 14 / 15

Thank you for your attention! References: 1 J. van den Berg, O. Häggstöm and J. Kahn. Some conditional correlation inequalities for percolation and related processes.

15 Thank you for your attention! References: 1 J. van den Berg, O. Häggstöm and J. Kahn. Some conditional correlation inequalities for percolation and related processes. Random Structures & Algorithms (2006). 2 T. Liggett and J. Steif. Stochastic domination: the contact process, Ising models and FKG measures. Ann. Inst. H Poin. (2006). 3 T. Liggett. Conditional association and spin systems. ALEA (2006). 4 J. van den Berg, S. Bethuelsen. Stochastic domination in space-time for the contact process. Accepted by Random Structures & Algorithms (2017). (S.A. Bethuelsen) 15 / 15

arxiv:math/ v1 [math.pr] 10 Jun 2004

arxiv:math/ v1 [math.pr] 10 Jun 2004 Slab Percolation and Phase Transitions for the Ising Model arxiv:math/0406215v1 [math.pr] 10 Jun 2004 Emilio De Santis, Rossella Micieli Dipartimento di Matematica Guido Castelnuovo, Università di Roma