Stretched Exponential Fixation in Stochastic Ising Models at Zero Temperature

Fontes, Luiz Renato; Schonmann, Roberto H.; Sidoravičius, Vladas

doi:10.1007/s002200200658

Cited by 103 publications

(139 citation statements)

References 15 publications

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“…Similar stochastic processes on different types of lattices have been studied in various papers. See, for example, [3,8,10,15,16,17,18] for models on Z d and [12] for a model on the homogeneous tree of degree three. Such models are also discussed extensively in the physics literature, usually on Z d (see, for example, [7] and [14]).…”

Section: Definition Of the Model(s) And Resultsmentioning

confidence: 99%

Cardy’s formula for some dependent percolation models

Camia¹,

Newman

Sidoravičius

2002

Bull Braz Math Soc

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We prove Cardy's formula for rectangular crossing probabilities in dependent site percolation models that arise from a deterministic cellular automaton with a random initial state. The cellular automaton corresponds to the zero-temperature case of Domany's stochastic Ising ferromagnet on the hexagonal lattice H (with alternating updates of two sublattices) [7]; it may also be realized on the triangular lattice T with flips when a site disagrees with six, five and sometimes four of its six neighbors.

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Section: Definition Of the Model(s) And Resultsmentioning

confidence: 99%

Cardy’s formula for some dependent percolation models

Camia¹,

Newman

Sidoravičius

2002

Bull Braz Math Soc

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“…(As a consequence of Schonmann's proof in [33], we have p c ([n] d , d) = o(1) if d log * n.) Morris [28] also used the techniques of [5] to prove that, in zero-temperature Glauber dynamics on Z d , the critical threshold for fixation converges to 1/2 as d → ∞ (see also [22]). …”

Section: Introductionmentioning

confidence: 99%

Bootstrap Percolation in High Dimensions

Balogh¹,

Bollobás²,

Morris³

2010

Combinator. Probab. Comp.

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Abstract. In r-neighbour bootstrap percolation on a graph G, a set of initially infected vertices A ⊂ V (G) is chosen independently at random, with density p, and new vertices are subsequently infected if they have at least r infected neighbours. The set A is said to percolate if eventually all vertices are infected. Our aim is to understand this process on the grid, [n] d , for arbitrary functions n = n(t), d = d(t) and r = r(t), as t → ∞. The main question is to determine the critical probability p c ([n] d , r) at which percolation becomes likely, and to give bounds on the size of the critical window. In this paper we study this problem when r = 2, for all functions n and d satisfying d log n. The bootstrap process has been extensively studied on [n] d when d is a fixed constant and 2 r d, and in these cases p c ([n] d , r) has recently been determined up to a factor of 1 + o(1) as n → ∞. At the other end of the scale, Balogh and Bollobás determined p c ([2] d , 2) up to a constant factor, and Balogh, Bollobás and Morris determined2+ε , and gave much sharper bounds for the hypercube. Here we prove the following result: let λ be the smallest positive root of the equation

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“…Similar models on different lattices have been studied in various papers; see, for example, [11,12,13,14,15,16,17] for models on Z d and [18] for a model on the homogeneous tree of degree three. Such models are also discussed extensively in the physics literature, usually on Z d (see, for example, [2] and [19]).…”

Section: Introductionmentioning

confidence: 99%

The Percolation Transition in the Zero-Temperature Domany Model

Camia

Newman²

2004

Journal of Statistical Physics

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We analyze a deterministic cellular automaton σ · = (σ n : n ≥ 0) corresponding to the zero-temperature case of Domany's stochastic Ising ferromagnet on the hexagonal lattice H. The state space S H = {−1, +1} H consists of assignments of −1 or +1 to each site of H and the initial state σ 0 = {σ 0 x } x∈H is chosen randomly with P (σ 0The sites of H are partitioned in two sets A and B so that all the neighbors of a site x in A belong to B and vice versa, and the discrete time dynamics is such that the σ · x 's with x ∈ A (respectively, B) are updated simultaneously at odd (resp., even) times, making σ · x agree with the majority of its three neighbors.In [1] it was proved that there is a percolation transition at p = 1/2 in the percolation models defined by σ n , for all times n ∈ [1, ∞]. In this paper, we study the nature of that transition and prove that the critical exponents β, ν and η of the dependent percolation models defined by σ n , n ∈ [1, ∞], have the same values as for standard two-dimensional independent site percolation (on the triangular lattice).

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Stretched Exponential Fixation in Stochastic Ising Models at Zero Temperature

Cited by 103 publications

References 15 publications

Cardy’s formula for some dependent percolation models

Cardy’s formula for some dependent percolation models

Bootstrap Percolation in High Dimensions

The Percolation Transition in the Zero-Temperature Domany Model

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