Gibbs States on Countable Sets

Preston, Christopher J.

doi:10.1017/cbo9780511897122

Cited by 359 publications

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“…The phase-transition with respect to the uniqueness of the Gibbs distribution occurs at the inverse-temperature β u = arctanh(1/b), as established in 1974 by Preston [34].…”

Section: Introductionmentioning

confidence: 88%

Mixing Time of Critical Ising Model on Trees is Polynomial in the Height

Ding

Lubetzky

Peres

2010

Commun. Math. Phys.

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Abstract:In the heat-bath Glauber dynamics for the Ising model on the lattice, physicists believe that the spectral gap of the continuous-time chain exhibits the following behavior. For some critical inverse-temperature β c , the inverse-gap is O(1) for β < β c , polynomial in the surface area for β = β c and exponential in it for β > β c . This has been proved for Z 2 except at criticality. So far, the only underlying geometry where the critical behavior has been confirmed is the complete graph. Recently, the dynamics for the Ising model on a regular tree, also known as the Bethe lattice, has been intensively studied. The facts that the inverse-gap is bounded for β < β c and exponential for β > β c were established, where β c is the critical spin-glass parameter, and the tree-height h plays the role of the surface area.In this work, we complete the picture for the inverse-gap of the Ising model on the b-ary tree, by showing that it is indeed polynomial in h at criticality. The degree of our polynomial bound does not depend on b, and furthermore, this result holds under any boundary condition. We also obtain analogous bounds for the mixing-time of the chain. In addition, we study the near critical behavior, and show that for β > β c , the inverse-gap and mixing-time are both exp[ ((β − β c )h)].

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“…The phase-transition with respect to the uniqueness of the Gibbs distribution occurs at the inverse-temperature β u = arctanh(1/b), as established in 1974 by Preston [34].…”

Section: Introductionmentioning

confidence: 88%

Mixing Time of Critical Ising Model on Trees is Polynomial in the Height

Ding

Lubetzky

Peres

2010

Commun. Math. Phys.

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“…Now let µ ∈ S(Λ, Φ) be a probability measure defined on Ω such that µ(σ) > 0 for any cell σ ∈ Ω; i.e µ is a Gibbs measure with some potential [12], [14]. The heredity coefficients p σ 1 σ 2 ,σ are defined as…”

Section: Construction Of Qsomentioning

confidence: 99%

On Non-Volterra Quadratic Stochastic Operators Generated by a Product Measure

Rozikov

Shamsiddinov

2009

Stochastic Analysis and Applications

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In this paper we describe a wide class of non-Volterra quadratic stochastic operators using N. Ganikhadjaev's construction of quadratic stochastic operators. By the construction these operators depend on a probability measure µ being defined on the set of all configurations which are given on a graph G. We show that if µ is the product of probability measures being defined on each maximal connected subgraphs of G then corresponding non-Volterra operator can be reduced to m number (where m is the number of maximal connected subgraphs of G) of Volterra operators defined on the maximal connected subgraphs. Our result allows to study a wide class of non-Volterra operators in the framework of the well known theory of Volterra quadratic stochastic operators.

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“…The Markov property depends on the time reversibility and irreducibility of the process (Preston 1974, p. 13). It then follows that (Preston 1974, p. 14):…”

Section: Spatial Temporal Processes and Interaction Modelsmentioning

confidence: 93%

“…The following discussion of binary interaction models draws on monographs and papers by Biggs (1976), Preston (1974), and Besag (1974). The intention is to link these three studies in order to develop the practical application of interaction models and their substantive interpretation.…”

Section: Interaction Modelsmentioning

confidence: 99%

Interaction Models and Spatial Diffusion Processes

Haining

1982

Geographical Analysis

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Suppose we have the following structure. Within a region, R, there are a finite number of sites or vertices. At each of these an observation is taken. Suppose for present purposes the observation is binary (0 or 1). The set of observations, across all the vertices, is called a configuration. The intention here is to discuss probability models for describing such configurations and in particular those configurations where independence across the set of vertices cannot be assumed.The relevant class of models from the mathematical and statistical literature is the class of interaction models. The geographers interest in these models may be traced back to the early literature on spatial autocorrelation and more generally stochastic univariate spatial modeling. However the important links between geographical problems and this formal structure have never been explicitly discussed.The purpose here is to discuss the class of interaction models with particular reference to spatial diffusion modeling. (The binary nature of the data may reflect a twefold classification of individuals into adopters or nonadopters but more general models do exist.) In the first section there is a fairly informal discussion of the structure of interaction models. The second section considers spatial temporal processes that yield specific interaction models at equilibrium. The remainder of the paper treats the relationship between interaction models and diffusion processes with, finally, an empirical application. The example is not ideal but may help to indicate how things might be done in practice.The term diffusion has a,rather particular meaning in this paper. It does not, for example, refer to outward spread from a core region. In fact it is better described as a process in which a particular attribute moves through a population. Each individual (site) either possesses or does not possess the attribute (is occupied or vacant). Individuals cycle between these two states in response to the local configuration of possession or nonpossession. The models may be suitable for describing certain types of epidemic processes or even innovation adoption. They may also be appropriate for the study of spatial price variation. Perhaps of greater importance at this stage though is to recognize the utility of this class of models for handling the spatial properties of limited interaction and limited dispersal (see sec.3).

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Gibbs States on Countable Sets

Cited by 359 publications

References 0 publications

Mixing Time of Critical Ising Model on Trees is Polynomial in the Height

Mixing Time of Critical Ising Model on Trees is Polynomial in the Height

On Non-Volterra Quadratic Stochastic Operators Generated by a Product Measure

Interaction Models and Spatial Diffusion Processes

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