Perfect simulation for the infinite random cluster model, Ising and Potts models at low or high temperature

Santis, Emilio De; Maffei, Andrea

doi:10.1007/s00440-014-0608-2

Cited by 5 publications

(8 citation statements)

References 16 publications

(35 reference statements)

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“…The independence of the decrements of the random walk ensures that, with probability 1, a sub-walk in B will appear infinitely many times. Once T (0) reaches the site V (0) r , the value X r,w 0 can be obtained from w V (0) r by using the recursion in (6), with an exception for each of the sub-walks in B identified before. Suppose that one of these sub-walks joins s = T (0)…”

Section: Resultsmentioning

confidence: 99%

“…5. If such a subpath appears connecting, say, the sites u and l (with u − l = n 0 ), extract U * uniform in (0, 1), independent of all the variables generated previously: if U * < ε, set X u = 1, compute by forward simulation X m , using (6) and the coupling functions f * (U * ; •) on the previously located segments (l, u], together with X n , n ∈ Λ for all n such that T (n) hits m, delete m from V and go to 2.;…”

Section: A Results With G Countablementioning

confidence: 99%

“…For perfect simulation in the multi-dimensional case the reader is addressed to e.g. [5,7,15]; also the continuity assumption can be relaxed, as in [6].…”

Section: Introduction and Main Definitionsmentioning

confidence: 99%

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One-Dimensional Infinite Memory Imitation Models with Noise

Santis

Piccioni

2015

J Stat Phys

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In this paper we study stochastic process indexed by Z constructed from certain transition kernels depending on the whole past. These kernels prescribe that, at any time, the current state is selected by looking only at a previous random instant. We characterize uniqueness in terms of simple concepts concerning families of stochastic matrices, generalizing the results previously obtained in De Santis and Piccioni (J. Stat. Phys., 150(6):1017-1029, 2013).

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Section: Resultsmentioning

confidence: 99%

Section: A Results With G Countablementioning

confidence: 99%

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One-Dimensional Infinite Memory Imitation Models with Noise

Santis

Piccioni

2015

J Stat Phys

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“…We end by briefly discussing the algorithmic aspects of our results. There is an extensive literature focused on perfect simulations of infinite-range models [4,6,12,13]; for an example involving the "high noise" regimes of the random-cluster model on Z d , see [5]. The proofs given in this paper rely on the method of coupling-from-the-past of Propp and Wilson [26].…”

Section: Introductionmentioning

confidence: 99%

Finitary codings for the random-cluster model and other infinite-range monotone models

Harel¹,

Spinka²

2018

Preprint

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Let G be a quasi-transitive graph on V. A random field X = (Xv) v∈V whose distribution is invariant under all automorphisms of G is said to be a factor of i.i.d. if there exists an i.i.d. process Y = (Yv) v∈V and an equivariant map ϕ such that ϕ(Y ) has the same distribution as X. Such a map, also called a coding, is said to be finitary if, for every v ∈ V, there exists a finite (but random) set U ⊂ V such that ϕ(Y )v is determined by {Yu} u∈U . We construct a coding for the random-cluster model on general quasi-transitive graphs, and show that the coding is finitary whenever the free and wired measures coincide. This strengthens a result of Häggström-Jonasson-Lyons [18]. We also prove that the coding radius has exponential tails in the sub-critical regime. As a corollary, we obtain a finitary coding for the sub-critical Potts model on G whose coding radius has exponential tails. In the case of G = Z d , we also construct a finitary, translation-equivariant coding for the sub-critical random-cluster and Potts models using a finite-valued i.i.d. process Y . To do this, we extend a mixing-time result of to infinite-range monotone models on quasi-transitive graphs of sub-exponential growth. Our methods also apply to any monotone model satisfying mild technical (but natural) requirements.

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“…There are important results for the convergence of the system to the Gibbs state in the case of low temperature (see e.g. [6,7,9,20]) or high temperature (see e.g. [6,8,19,21]).…”

Section: Introductionmentioning

confidence: 99%

Stochastic Ising model with flipping sets of spins and fast decreasing temperature

Cerqueti¹,

Santis²

2018

Ann. Inst. H. Poincaré Probab. Statist.

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This paper deals with the stochastic Ising model with a temperature shrinking to zero as time goes to infinity. A generalization of the Glauber dynamics is considered, on the basis of the existence of simultaneous flips of some spins. Such dynamics act on a wide class of graphs which are periodic and embedded in R d . The interactions between couples of spins are assumed to be quenched i.i.d. random variables following a Bernoulli distribution with support {−1, +1}. The specific problem here analyzed concerns the assessment of how often (finitely or infinitely many times, almost surely) a given spin flips. Adopting the classification proposed in [14], we present conditions in order to have models of type F (any spin flips finitely many times), I (any spin flips infinitely many times) and M (a mixed case). Several examples are provided in all dimensions and for different cases of graphs. The most part of the obtained results holds true for the case of zero-temperature and some of them for the cubic lattice L d = (Z d , E d ) as well.

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Perfect simulation for the infinite random cluster model, Ising and Potts models at low or high temperature

Abstract: In this article we create a new algorithm for the perfect simulation of the infinite random cluster model for a sufficiently small or a sufficiently high value of the parameters. This implies the simulation of the Ising and Potts models with free boundary conditions

Cited by 5 publications

References 16 publications

One-Dimensional Infinite Memory Imitation Models with Noise

One-Dimensional Infinite Memory Imitation Models with Noise

Finitary codings for the random-cluster model and other infinite-range monotone models

Stochastic Ising model with flipping sets of spins and fast decreasing temperature

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