Pierre Picco

On the Existence of Thermodynamics for the Random Energy Model

Derridas random energy model is considered. Almost sure andLP convergence of the free energy at any inverse temperature β are proven. Rigorous upper and lower bounds to the finite size corrections to the free energy are given.

Gibbs states of the Hopfield model in the regime of perfect memory

SummaryWe study the thermodynamic properties of the Hopfield model of an autoassociative memory. IfN denotes the number of neurons andM (N) the number of stored patterns, we prove the following results: IfM/N↓ 0 asN↑ ∞, then there exists an infinite number of infinite volume Gibbs measures for all temperaturesT<1 concentrated on spin configurations that have overlap with exactly one specific pattern. Moreover, the measures induced on the overlap parameters are Dirac measures concentrated on a single point and the Gibbs measures on spin configurations are products of Bernoulli measures. IfM/N → α, asN↓∞ for α small enough, we show that for temperaturesT smaller than someT(α)<1, the induced measures can have support only on a disjoint union of balls around the previous points, but we cannot construct the infinite volume measures through convergent sequences of measures.

Mathematical aspects of spin glasses and neural networks

Statics mean field models Bovier and V. Gayard Hopfield models as generalized random mean field models comets - the Martingale method for mean field disordered systems at high temperature.

Gibbs states of the Hopfield model with extensively many patterns

We consider the Hopfield model withM(N)=αN patterns, whereN is the number of neurons. We show that if α is sufficiently small and the temperature sufficiently low, then there exist disjoint Gibbs states for each of the stored patterns, almost surely with respect to the distribution of the random patterns. This solves a provlem left open in previous work. The key new ingredient is a self-averaging result on the free energy functional. This result has considerable additional interest and some consequences are discussed. A similar result for the free energy of the Sherrington-Kirkpatrick model is also given.

Cluster expansion ford-dimensional lattice systems and finite-volume factorization properties

We consider classical lattice systems with finite-range interactions ind dimensions. By means of a block-decimation procedure, we transform our original system into a polymer system whose activity is small provided a suitable factorization property of finite-volume partition functions holds. In this way we extend a result of Olivieri.

EXISTENCE OF NON-TRIVIAL QUASI-STATIONARY DISTRIBUTIONS IN THE BIRTH-DEATH CHAIN

We study conditions for the existence of non-trivial quasi-stationary distributions for the birth-and-death chain with 0 as absorbing state. We reduce our problem to a continued fractions one that can be solved by using extensions of classical results of this theory. We also prove that there exist normalized quasi-stationary distributions if and only if 0 is geometrically absorbing.

Fluctuations in Derrida's Random Energy and Generalized Random Energy Models

The fluctuations of the finite-size corrections to the free energy per site of the random energy model (REM) and the generalized random energy model (GREM) are investigated. Almost sure behavior for the corrections of order (logN)/N is given. We also prove convergence in distribution for the corrections of order 1/N.

On the existence of thermodynamics for the generalized random energy model

Derridas generalized random energy model is considered. Almost sure andLp convergence of the free energy at any inverse temperatureβ are proven for an arbitrary numbern of hierarchical levels. The explicit form of the free energy is given in the most general case and the limitn→∞ is discussed.

METASTABILITY AND CONVERGENCE TO EQUILIBRIUM FOR THE RANDOM FIELD CURIE-WEISS MODEL

We study a dynamics for the magnetization of the random field Curie–Weiss model. A metastable behavior is exhibited and asymptotic estimates on the speed of convergence to equilibrium are given. The results are given almost surely and in law with respect to the realizations of the random magnetic fields.

Large deviation principles for the Hopfield model and the Kac-Hopfield model

SummaryWe study the Kac version of the Hopfield model and prove a Lebowitz-Penrose theorem for the distribution of the overlap parameters. At the same time, we prove a large deviation principle for the standard Hopfield model with infinitely many patterns.