Invariant Gibbs Measures Research Articles

Hausdorff Dimension of Sets of Generic Points for Gibbs Measures

For a translation invariant Gibbs measure ν on the configuration space X of a lattice finite spin system, we consider the set X ν of generic points. Using a Breiman type convergence theorem on the set X μ of generic points of an arbitrary translation invariant probability measure μ on X, we evaluate the Hausdorff dimension of the set X ν with respect to any metric out of a wide class of “scale” metrics on X (including Billingsley metrics generated by Gibbs measures).

Large Deviations and¶Variational Principle for Harmonic Crystals

We consider massless Gaussian fields with covariance related to the Green function of a long range random walk on ℤ d . These are viewed as Gibbs measures for a linear-quadratic interaction. We establish thermodynamic identities and prove a version of Gibbs' variational principle, showing that translation invariant Gibbs measures are characterized as minimizers of the relative entropy density. We then study the large deviations of the empirical field of a Gibbs measure. We show that a weak large deviation principle holds at the volume order, with rate given by the relative entropy density.

Graph Homomorphisms and Phase Transitions

We model physical systems with “hard constraints” by the space Hom(G, H) of homomorphisms from a locally finite graph G to a fixed finite constraint graph H. For any assignment λ of positive real activities to the nodes of H, there is at least one Gibbs measure on Hom(G, H); when G is infinite, there may be more than one. When G is a regular tree, the simple, invariant Gibbs measures on Hom(G, H) correspond to node-weighted branching random walks on H. We show that such walks exist for every H and λ, and characterize those H which, by admitting more than one such construction, exhibit phase transition behavior.

A subshift of finite type that is equivalent to the Ising model

AbstractFor the Ising model with rational parameters we show how to construct a subshift of finite type that is equivalent to this Ising model, in that the translation invariant Gibbs measures for the Ising model and the measures of maximal entropy for the subshift of finite type can be identified in a natural way. This is generalized to the non-translation invariant case as well. We also show how to construct, given any H > 0, an ergodic measure of maximal entropy for a subshift of finite type and a continuous factor, such that the factor has entropy H.

Large deviations for ℤ d -Actions

We establish large deviations bounds for translation invariant Gibbs measures of multidimensional subshifts of finite type. This generalizes [FO] and partially [C, O, and B], where only full shifts were considered. Our framework includes, in particular, the hard-core lattice gas models which are outside of the scope of [FO, C, O, and B].

Mesures de Gibbs invariantes et mesures d'equilibre

We are interested here in the characterization on a symbolic space, of invariant Gibbs measures as equilibrium measures. The first result in this topic was obtained by Lanford and Ruelle (see for example [6]). This problem involves different objects that can all be defined by using the only amenability of the translation group and the only continuity of the local specification. We therefore tried to state our theorems in this general frame. Among the elements of our proof, there is the use of the information gain introduced by H. Follmer [1] and some arguments similar to those of C.J. Preston in [5]. But the amenability techniques that we widely develop in [2], [4] and [7] are decisive tools for getting the result. The corresponding problem for subshifts is not considered in the present paper so symbolic spaces are product spaces.

On invariant Gibbs measures for the generalized KdV equations

We consider the defocusing generalized KdV equations on the circle. In particular, we construct global-in-time solutions with initial data distributed according to the Gibbs measure and show that the law of the random solutions, at any time, is again given by the Gibbs measure. In handling a nonlinearity of an arbitrary high degree, we make use of the Hermite polynomials and the white noise functional.