Jean-René Chazottes

Projection of Markov Measures May Be Gibbsian

We study the induced measure obtained from a 1-step Markov measure, supported by a topological Markov chain, after the mapping of the original alphabet onto another one. We give sufficient conditions for the induced measure to be a Gibbs measure (in the sense of Bowen) when the factor system is again a topological Markov chain. This amounts to constructing, when it does exist, the induced potential and proving its Hölder continuity. This is achieved through a matrix method. We provide examples and counterexamples to illustrate our results.

Optimal concentration inequalities for dynamical systems

For dynamical systems modeled by a Young tower with exponential tails, we prove an exponential concentration inequality for all separately Lipschitz observables of n variables. When tails are polynomial, we prove polynomial concentration inequalities. Those inequalities are optimal. We give some applications of such inequalities to specific systems and specific observables.

Exponential Distribution for the Occurrence of Rare Patterns in Gibbsian Random Fields

We study the distribution of the occurrence of rare patterns in sufficiently mixing Gibbs random fields on the lattice ℤd, d≥2. A typical example is the high temperature Ising model. This distribution is shown to converge to an exponential law as the size of the pattern diverges. Our analysis not only provides this convergence but also establishes a precise estimate of the distance between the exponential law and the distribution of the occurrence of finite patterns. A similar result holds for the repetition of a rare pattern. We apply these results to the fluctuation properties of occurrence and repetition of patterns: We prove a central limit theorem and a large deviation principle.

Fluctuations of Observables in Dynamical Systems: From Limit Theorems to Concentration Inequalities

We start by reviewing recent probabilistic results on ergodic sums in a large class of (nonuniformly) hyperbolic dynamical systems. Namely, we describe the central limit theorem, the almost-sure convergence to the Gaussian and other stable laws, and large deviations.

On Concentration Inequalities and Their Applications for Gibbs Measures in Lattice Systems

We consider Gibbs measures on the configuration space

On the Zero-Temperature Limit of Gibbs States

S^{{\mathbb {Z}}^d}

Poisson approximation for the number of visits to balls in non-uniformly hyperbolic dynamical systems

SZd, where mostly