Véronique Gayrard

Aix-Marseille University | AMU · Institut de Mathématiques de Marseille, UMR 7373

We study the SK model at inverse temperature \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\beta >0$$\end{document} and strictly positive field \documentclass[12pt]{m...

We study the SK model at inverse temperature $\beta>0$ and strictly positive field $h>0$ in the region of $(\beta,h)$ where the replica-symmetric formula is valid. An integral representation of the partition function derived from the Hubbard-Stratonovitch transformation combined with a duality formula is used to prove that the infinite volume free...

By studying the two-time overlap correlation function, we give a comprehensive analysis of the phase diagram of the Random Hopping Dynamics of the Random Energy Model (REM) on time-scales that are exponential in the volume. These results are derived from the convergence properties of the clock process associated to the dynamics and fine properties...

We study the aging behavior of the Random Energy Model (REM) evolving under Metropolis dynamics. We prove that a classical two-time correlation function converges almost surely to the arcsine law distribution function that characterizes activated aging, as predicted in the physics literature, in the optimal domain of the time-scale and temperature...

By studying the two-time overlap correlation function, we give a comprehensive analysis of the phase diagram of the Random Hopping Dynamics of the Random Energy Model (REM) on time-scales that are exponential in the volume. These results are derived from the convergence properties of the clock process associated to the dynamics and fine properties...

These proceedings of the conference Advances in Statistical Mechanics, held in Marseille, France, August 2018, focus on fundamental issues of equilibrium and non-equilibrium dynamics for classical mechanical systems, as well as on open problems in statistical mechanics related to probability, mathematical physics, computer science, and biology. S...

We derive scaling limit results for the Random Hopping Dynamics for the cascading two-level GREM at low temperature at extreme time scales. It is known that in the cascading regime there are two static critical temperatures. We show that there exists a (narrow) set of fine tuning temperatures; when they lie below the static lowest critical temperat...

The GREM-like trap model is a continuous time Markov jump process on the leaves of a finite volume $L$-level tree whose transition rates depend on a trapping landscape built on the vertices of the whole tree. We prove that the natural two-time correlation function of the dynamics ages in the infinite volume limit and identify the limiting function....

This volume presents five different methods recently developed to tackle the large scale behavior of highly correlated random systems, such as spin glasses, random polymers, local times and loop soups and random matrices. These methods, presented in a series of lectures delivered within the Jean-Morlet initiative (Spring 2013), play a fundamental r...

We study the aging behavior of a truncated version of the Random Energy Model evolving under Metropolis dynamics. We prove that the natural time-time correlation function defined through the overlap function converges to an arcsine law distribution function, almost surely in the random environment and in the full range of time scales and temperatur...

This paper extends recent results on ageing in mean field spin glasses on short time scales, obtained by Ben Arous and Gün (Commun Pure Appl Math 65:77-127, 2012) in law with respect to the environment, to results that hold almost surely, respectively in probability, with respect to the environment. It is based on the methods put forward in (Gayrar...

Using a method developed by Durrett and Resnick, [23], we establish general criteria for the convergence of properly rescaled clock processes of random dynamics in random environments on infinite graphs. This extends the results of Gayrard, [27], Bovier and Gayrard, [20], and Bovier, Gayrard, and Svejda, [21], and gives a unified framework for prov...

We derive a general criterion for the convergence of clock processes in random dynamics in random environments that is applicable in cases when correlations are not negligible, extending recent results by Gayrard [15,16], based on general criterion for convergence of sums of dependent random variables due to Durrett and Resnick [13]. We demonstrate...

In this paper the celebrated arcsine aging scheme of Ben Arous and ̌Cerný is taken up. Using a brand new approach based on point processes and weak convergence techniques, this scheme is implemented in a broad class of Markov jump processes in random environments that includes Glauber dynamics of discrete disordered systems. More specifically, cond...

We obtain scaling limit results for asymmetric trap models and their infinite volume counterparts, namely asymmetric K processes. Aging results for the latter processes are derived therefrom.

We introduce trap models on a finite volume k-level tree as a class of Markov jump processes with state space the leaves of that tree. They serve to describe the GREM-like trap model of Sasaki-Nemoto. Under suitable conditions on the parameters of the trap model, we establish its infinite volume limit, given by what we call a K process in an infini...

Applying the new tools developed in [G1], we investigate the arcsine aging regime of the random hopping time dynamics of the REM. Our results are optimal in several ways. They cover the full time-scale and temperature domain where this phenomenon occurs. On this domain the limiting clock process and associated time correlation function are explicit...

In this paper the celebrated arcsine aging scheme of G. Ben Arous and J. Cern\'y is taken up. Using a brand new approach based on point processes and weak convergence techniques, this scheme is implemented in a wide class of Markov processes that can best be described as Glauber dynamics of discrete disordered systems. More specifically, conditions...

We study the simple random walk on the n-dimensional hypercube, in particular its hitting times of large (possibly random) sets. We give simple conditions on these sets ensuring that the properly rescaled hitting time is asymptotically exponentially distributed, uniformly in the starting position of the walk. These conditions are then verified for...

We introduce here a new universality conjecture for levels of random Hamiltonians, in the same spirit as the local REM conjecture made by S. Mertens and H. Bauke. We establish our conjecture for a wide class of Gaussian and non-Gaussian Hamiltonians, which include the $p$-spin models, the Sherrington-Kirkpatrick model and the number partitioning pr...

This work addresses potential theoretic questions for the standard nearest neighbor random walk on the hypercube $\{-1,+1\}^N$. For a large class of subsets $A\subset\{-1,+1\}^N$ we give precise estimates for the harmonic measure of $A$, the mean hitting time of $A$, and the Laplace transform of this hitting time. In particular, we give precise suf...

We continue the analysis of the problem of metastability for reversible diffusion processes, initiated in [BEGK3], with a precise analysis of the low-lying spectrum of the generator. Recall that we are considering processes with generators of the form +rF ( )r on R or subsets of R , where F is a smooth function with finitely many local minima. Here...

We develop a potential theoretic approach to the problem of metastability for reversible diffusion processes with generators of the form +rF ( )r on R or subsets of , where F is a smooth function with finitely many local minima. In analogy to previous work in discrete Markov chains, we show that metastable exit times from the attractive domains of...

We investigate the long-time behavior of the Glauber dynamics for the random energy model below the critical temperature. We give very precise estimates on the motion of the process to and between the states of extremal energies. We show that when disregarding time, the consecutive steps of the process on these states are governed by a Markov chain...

We investigate the long-time behavior of the Glauber dynamics for the random energy model below the critical temperature. We establish that for a suitably chosen timescale that diverges with the size of the system, one can prove that a natural autocorrelation function exhibits aging. Moreover, we show that the long-time asymptotics of this function...

We study the interface between liquid and vapor in the context of the van der Waals theory, considering the non-local free energy functional recently derived by Lebowitz, Mazel, and Presutti from a system of particles in the continuum with Kac potentials. We prove that the density profile between vapor and liquid is monotone when the inverse temper...

We study a large class of reversible Markov chains with discrete state space and transition matrix P N . We define the notion of a set of metastable points as a subset of the state space Γ N such that (i) this set is reached from any point x∈Γ N without return to x with probability at least b N , while (ii) for any two points x, y in the me...

The random energy model (REM) has become a key reference model for glassy systems. In particular, it is expected to provide a prime example of a system whose dynamics shows aging, a universal phenomenon characterizing the dynamics of complex systems. The analysis of its activated dynamics is based on so-called trap models, introduced by Bouchaud, t...

We study a class of Markov chains that describe reversible stochastic dynamics of a large class of disordered mean field models at low temperatures. Our main purpose is to give a precise relation between the metastable time scales in the problem to the properties of the rate functions of the corresponding Gibbs measures. We derive the analog of the...

In this Letter we announce rigorous results that elucidate the relation between metastable states and low-lying eigenvalues in Markov chains in a much more general setting and with considerably greater precision than has so far been available. This includes a sharp uncertainty principle relating all low-lying eigenvalues to mean times of metastabl...

We prove a large deviation principle on path space for a class of discrete time Markov processes whose state space is the intersection of a regular domain $\L\subset \R^d$ with some lattice of spacing $\e$. Transitions from $x$ to $y$ are allowed if $\e^{-1}(x-y)\in \D$ for some fixed set of vectors $\D$. The transition probabilities $p_\e(t,x,y)$,...

We study the finite dimensional marginals of the Gibbs measure in the Hopfield model at low temperature when the number of patterns, M , is proportional to the volume with a sufficiently small proportionality constant ff ? 0. It is shown that even when a single pattern is selected (by a magnetic field or by conditioning), the marginals do not conve...

Mean field models, random or not, are very important for explaining simply the general phenomenon of phase transitions. However, for random systems, in general, their analysis, as many of the contributions in this volume confirm, is not simple at all, a fact which may justify the amount of effort spent on them. In spite of all that, mean fields mod...

We prove a central limit theorem for the finite dimensional marginals of the Gibbs distribution of the macroscopic `overlap'-parameters in the Hopfield model in the case where the number of random `patterns', M , as a function of the system size N satisfies lim N"1 M(N)=N = 0, without any assumptions on the speed of convergence. The covariance matr...

: We survey the statistical mechanics approach to the analysis of neural networks of the Hopfield type. We consider both models on complete graphs (mean-field), random graphs (dilute model), and on regular lattices (Kac-model). We try to explain the main ideas and techniques, as well as the results obtained by them, without however going into too m...

We study a one-dimensional version of the Hopfield model with long, but finite range interactions below the critical temperature. In the thermodynamic limit we obtain large deviation estimates for the distribution of the ``local'' overlaps, the range of the interaction, $\gamma^{-1}$, being the large parameter. We show in particular that the local...

We give a comprehensive self-contained review on the rigorous analysis of the thermodynamics of a class of random spin systems of mean field type whose most prominent example is the Hopfield model. We focus on the low temperature phase and the analysis of the Gibbs measures with large deviation techniques. There is a very detailed and complete pict...

We give a comprehensive self-contained review on the rigorous analysis of the thermodynamics of a class of random spin systems of mean field type whose most prominent example is the Hopfield model. We focus on the low temperature phase and the analysis of the Gibbs measures with large deviation techniques. There is a very detailed and complete pict...

Standard large deviation estimates or the use of the Hubbard– Stratonovich transformation reduce the analysis of the distribution of the over- lap parameters essentially to that of an explicitly known random function N; ÿ on RM . In this article we present a rather careful study of the structure of the minima of this random function related to the...

We prove a large deviation principle for the finite dimensional marginals of the Gibbs distribution of the macroscopic `overlap'-parameters in the Hopfield model in the case where the number of random patterns, $M$, as a function of the system size $N$ satisfies $\limsup M(N)/N=0$. In this case the rate function (or free energy as a function of the...

We 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.

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

We consider the Hopfield model withM(N)=N patterns, whereN is the number of neurons. We show that if 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 wo...

We study the thermodynamic properties of the Hopfield model of an autoassociative memory. If N denotes the number of neurons and M(N) the number of stored patterns, we prove the following results: If M/N↓0 as N↑∞, then there exists an infinite number of infinite volume Gibbs measures for all temperatures T<1 concentrated on spin configurations that...

: We review some recent rigorous results in the theory of neural networks, and in particular on the thermodynamic properties of the Hopfield model. In this context, the model is treated as a Curie-Weiss model with random interactions and large deviation techniques are applied. The tractability of the random interactions depends strongly on how the...

We study the Curie-Weiss version of an Ising spin system with random, positively biased couplings. In particular, the case where the couplings ij take the values one with probabilityp and zero with probability 1 –p, which describes the Ising model on a random graph, is considered. We prove that ifp is allowed to decrease with the system sizeN in su...

We study the Hopfield model of an autoassociative memory on a random graph onN vertices where the probability of two vertices being joined by a link isp(N). Assuming thatp(N) goes to zero more slowly thanO(1/N), we prove the following results: (1) If the number of stored patternsm(N) is small enough such thatm(N)/Np(N) 0, asN, then the free energy...

A neural network is defined on a random graph describing the a priori dendritic connectivity of the N neurons. Using the techniques of Newman, we present a rigorous analysis of the energy landscape for the Hopfield Hamiltonian on such a network. It is found that if the probability, p, that two given neurons are connected is at least as big as \( \s...

We prove the almost sure convergence of the free energy and of the overlap order parameters in aq-state version of the Hopfield neural network model. We compute explicitly these limits for all temperatures different from some critical value. The number of stored patterns is allowed to grow with the size of the systemN like (/lnq) lnN. We study the...

We study a neural network model consisting ofN neurons where a dendritic connection between each pair of neurons exists with probabilityp and is absent with probability 1-p. For the Hopfield Hamiltonian on such a network, we prove that ifp c[(lnN)/N]1/2, the model can store at leastm= cpN patterns, where c 0.027 ifc 3 and decreases proportional to...

We study a class of Markov chains that describe reversible stochastic dynamics of a large class of disordered mean field models at low temperatures. Our main purpose is to give a precise relation between the metastable time scales in the problem to the properties of the rate functions of the corresponding Gibbs measures. We derive the analog of the...