Arnaud Le Ny

Gibbsian Description of Mean-Field Models

We introduce a new framework to describe mean-field models in the spirit of the DLR description of probability measures on infinite prodnet probability spaces used for lattice spin systems. The approach, originally introduced by C. Kuelske in 2003, is inspired by the generalized Gibbsian formalism recently developed in the context of the Dobrushin program of restoration of Gibbsianness, and enables the recovery of many of its features in the mean-field context. It is based on a careful study of the continuity properties of the limiting conditional probabilities of the finite-volume mean-field measures as a function of, empirical averages, when the limiting procedure is properly done to avoid trivialities. This contribution is an extended version of a poster presented at the Xth Brazilian school of probability and has mainly a review character.

Entropic Repulsion and Lack of the g-Measure Property for Dyson Models

We consider Dyson models, Ising models with slow polynomial decay, at low temperature and show that its Gibbs measures deep in the phase transition region are not g-measures. The main ingredient in the proof is the occurrence of an entropic repulsion effect, which follows from the mesoscopic stability of a (single-point) interface for these long-range models in the phase transition region.

Almost Gibbsianness and Parsimonious Description of the Decimated 2d-Ising Model

In this paper, we complete and provide details for the existing characterizations of the decimation of the Ising model on

Spin-Flip Dynamics of the Curie-Weiss Model: Loss of Gibbsianness with Possibly Broken Symmetry

\mathbb{Z}^{2}

Relative entropy and variational properties of generalized Gibbsian measures

in the generalized Gibbs context. We first recall a few features of the Dobrushin program of restoration of Gibbsianness and present the construction of global specifications consistent with the extremal decimated measures. We use them to prove that these renormalized measures are almost Gibbsian at any temperature and to analyse in detail its convex set of DLR measures. We also recall the weakly Gibbsian description and complete it using a potential that admits a quenched correlation decay, i.e. a well-defined configuration-dependent length beyond which this potential decays exponentially. We use these results to incorporate these decimated measures in the new framework of parsimonious random fields that has been recently developed to investigate probability aspects related to neurosciences.