Clusters and recurrence in the two-dimensional zero-temperature stochastic ising model

Camia, Federico; Santis, Emilio De; Newman, Charles M.

doi:10.1214/aoap/1026915616

Cited by 19 publications

(44 citation statements)

References 21 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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“…Many variants of this model have been studied in diverse settings and disciplines such as Economics (e.g., [5,1]), Statistical Mechanics (e.g., [10,6,4]), Computer Science (see a survey by Shah [18]), and Mathematics (e.g., [11,3]). These include variations on how the agents acquire their initial opinions (e.g., deterministically, at random, arbitrarily or through some other process), what they aim to achieve in this process (e.g., rational agents in Economics [7,1,16], message passing agents in Computer Science [12]), and how they go about updating their opinions in order to achieve this.…”

Section: Introductionmentioning

confidence: 99%

“…This is the smallest sigma-algebra for which X is measurable 3. We slightly weaken his definition; he requires that A i t = s for all i in some finite time t.…”

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confidence: 99%

Majority Dynamics and the Retention of Information

Tamuz

Tessler

2014

Isr. J. Math.

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We consider a group of agents connected by a social network who participate in majority dynamics: each agent starts with an opinion in {-1,+1} and repeatedly updates it to match the opinion of the majority of its neighbors. We assume that one of {-1,+1} is the "correct" opinion S, and consider a setting in which the initial opinions are independent conditioned on S, and biased towards it. They hence contain enough information to reconstruct S with high probability. We ask whether it is still possible to reconstruct S from the agents' opinions after many rounds of updates. While this is not the case in general, we show that indeed, for a large family of bounded degree graphs, information on S is retained by the process of majority dynamics. Our proof technique yields novel combinatorial results on majority dynamics on both finite and infinite graphs, with applications to zero temperature Ising models.Comment: 19 pages, accepted for publication in the Israel Journal of Mathematic

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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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Clusters and recurrence in the two-dimensional zero-temperature stochastic ising model

Cited by 19 publications

References 21 publications

Cardy’s formula for some dependent percolation models

Cardy’s formula for some dependent percolation models

Majority Dynamics and the Retention of Information

The Percolation Transition in the Zero-Temperature Domany Model

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