p-adic Gibbs measures and Markov random fields on countable graphs

The notions of the Gibbs measure and of the Markov random field are known to coincide in the real case. But in the p-adic case, the class of p-adic Markov random fields is broader than that of p-adic Gibbs measures. We construct p-adic Markov random fields (on finite graphs) that are not p-adic Gibbs measures. We define a p-adic Markov random field on countable graphs and show that the set of such fields is a nonempty closed subspace in the set of all p-adic probability measures     

Large deviations for Gibbs random fields

A large deviation principle for Gibbs random fields on Z^d is proven and a corresponding large deviations proof of the Gibbs variational formula is given. A generalization of the Lanford theory of large deviations is also obtained.This work was partially supported by NSF-DMR81-14726     

Symmetric Gibbs measures

We prove that certain Gibbs measures on subshifts of finite type are nonsingular and ergodic for certain countable equivalence relations, including the orbit relation of the adic transformation (the same as equality after a permutation of finitely many coordinates). The relations we consider are defined by cocycles taking values in groups, including some nonabelian ones. This generalizes (half of) the identification of the invariant ergodic probability measures for the Pascal adic transformation as exactly the Bernoulli measures-a version of de Finetti's theorem. Generalizing the other half, we characterize the measures on subshifts of finite type that are invariant under both the adic and the shift as the Gibbs measures whose potential functions depend on only a single coordinate. There are connections with and implications for exchangeability, ratio limit theorems for transient Markov chains, interval splitting procedures, `canonical' Gibbs states, and the triviality of remote sigma-fields finer than the usual tail field.

     

Transformations of Gibbs measures

We study local transformations of Gibbs measures. We establish sufficient conditions for the quasilocality of the images and obtain results on the existence and continuity properties of their relative energies. General results are illustrated by simple examples. Received: 11 November 1997 / Revised version: 20 February 1998     

Combinatorial criteria for uniqueness of Gibbs measures

We generalize previously known conditions for uniqueness of the Gibbs measure in statistical physics models by presenting conditions of any finite size for models on any underlying graph. We give two dual conditions, one requiring that the total influence on a site is small, and the other that the total influence of a site is small. Our proofs are combinatorial in nature and use tools from the analysis of discrete Markov chains, in particular the path coupling method. The implications of our conditions for the mixing time of natural Markov chains associated with the models are discussed as well. We also present some examples of models for which the conditions hold. © 2005 Wiley Periodicals, Inc. Random Struct. Alg., 2005     

Weakly coupled Gibbs measures

Summary We formulate an abstract functional-analytic framework for the study of Gibbs measures on infinite product spaces. Working in this frame-work, we present a detailed analysis of the weak-coupling regime. Specifically, we derive general theorems on existence of the Gibbs measure, analyticity in its component Gibbs factors, and exponential decay of correlations and truncated expectations in the spread of distant families of random variables. In translation-invariant situations we obtain a central limit theorem. Our main tool is a series expansion in truncated expectations, which we analyze with L ^p methods.Original title: Analyticity and Decay of Correlations in Weakly Coupled Lattice Models.Supported by N.S.F. Grant PHY76-17191Dedicated to Professor Leopold Schmetterer     

Gibbs and equilibrium measures for elliptic functions

Because of its double periodicity, each elliptic function canonically induces a holomorphic dynamical system on a punctured torus. We introduce on this torus a class of summable potentials. With each such potential associated is the corresponding transfer (Perron-Frobenius-Ruelle) operator. The existence and uniquenss of Gibbs states and equilibrium states of these potentials are proved. This is done by a careful analysis of the transfer operator which requires a good control of all inverse branches. As an application a version of Bowens formula for expanding elliptic maps is obtained.The research of the second author was supported in part by the NSF Grant DMS 0400481 and INT 0306004.     

Existence of Gibbs measures for countable Markov shifts

We prove that a potential with summable variations and finite pressure on a topologically mixing countable Markov shift has a Gibbs measure iff the transition matrix satisfies the big images and preimages property. This strengthens a result of D. Mauldin and M. Urbanski (2001) who showed that this condition is sufficient.

     

Periodic Gibbs measures for the Potts model in translation-invariant and periodic external fields on the Cayley tree

Theoretical and Mathematical Physics - We study the Potts model in translation-invariant and periodic external fields on the Cayley tree of order $${k\geq 2}$$ . For the Potts model in a...     

Duality transformations of Gibbs random fields

The duality of Gibbs fields are examined in three forms: the Poisson summation formula, “electrodynamic” representation, and Hamiltonian duality. New low-temperature expansions are obtained and systems with nonsummable interaction are studied.     

Gibbs measures on Brownian currents

Motivated by applications to quantum field theory, we consider Gibbs measures for which the reference measure is Wiener measure and the interaction is given by a double stochastic integral and a pinning external potential. In order to properly characterize these measures through Dobrushin‐Lanford‐Ruelle (DLR) equations, we are led to lifting Wiener measure and other objects to a space of configurations where the basic observables are not only the position of the particle at all times but also the work done by test vector fields. We prove existence and basic properties of such Gibbs measures in the weak coupling regime by means of cluster expansion. © 2008 Wiley Periodicals, Inc.     

Some new examples of Gibbs measures

We investigate Gibbs measures on general subshifts. In particular we show the uniqueness of Gibbs measures as equilibrium states and we construct such measures on other spaces than mixing subshifts of finite type or sofic systems.     

ABC methods for model choice in Gibbs random fields

We consider the problem of model selection within the class of Gibbs random fields. In a Bayesian framework, this choice relies on the evaluation of the posterior probabilities of all models. We define an extended parameter setting, including the model index and show the existence of a corresponding sufficient statistic made of the conjunction of the sufficient statistics of all models. We use this statistic to derive an ABC algorithm. To cite this article: A. Grelaud et al., C. R. Acad. Sci. Paris, Ser. I 347 (2009).     

Time reversal and stationary Gibbs measures

Markov chains on an infinite product space are considered whose transition kernel is of the Gibbsian type. It is proved that then a stationary probability measure is Gibbsian if and only if the transition kernel of the reversed chain is also Gibbsian.     

Construction of an uncountable number of limiting Gibbs measures in the inhomogeneous ising model

We prove the existence of an uncountable number of limiting Gibbs measures in the inhomogeneous Ising model on a Cayley tree and describe them constructively. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 118, No. 1, pp. 95–104, January, 1999.