Glauber dynamics of the random energy model. I: Metastable motion on the extreme states
type: Item
instance of: scholarly article
title: Glauber dynamics of the random energy model. I: Metastable motion on the extreme states
zbMATH Open document ID: 1037.82038
author: Gerard Ben Arous
author: Anton Bovier
author: V. Gayrard
published in: Communications in Mathematical Physics
publication date: 2003-08-17T00:00:00Z
review text: This paper is the first part of a rigorous study of the Glauber dynamics for the so-called Random Energy Model (REM). This mean-field model has been introduced as a paradigm to study spin-glass disordered systems, which complex behavior has been predicted by physicists. On the hypercube \(\mathcal{S}_N=\{-1,+1\}^N\), the random Gibbs measures associated to i.i.d. normal random variables \(X_\sigma, \sigma \in \mathcal{S}_N\) (with \(\mathbb{E}_\sigma= X_\sigma;v; 0\)) is defined at inverse temperature \(\beta\) via weights \(\mu_{\beta,N}(\sigma)\) proportional to a Boltzmann factor \(e^{\beta \sqrt{N} E_\sigma}\). A phase transition occurs at \(\beta_c=\sqrt{2 \ln{2}}\) with the following behavior : for \(\beta \leq \beta_c\), the Gibbs measures are asymptotically supported on the set of states \(\sigma\) such that \(\sigma\) is of the order of \(\sqrt{N} \beta\) and no single configuration has a positive mass, while in the uniqueness regime (\(\beta > \beta_c\)), the Gibbs measure gives positive mass to the extreme statistics of the family \(\sigma\) and the entire family of masses converges in a suitable sense to a random process called Ruelle's point process. Together with its companion paper [Commun. Math. Phys. 236, No. 1, 1--54 (2003; Zbl 1037.82035)], this paper studied precisely this low temperature regime in order to describe mathematically the occurrence of an aging phenomenon, as predicted by theoretical physicists. This description uses a discrete time Glauber dynamics of the REM : starting from an a priori (random) distribution of the configuration \(\sigma\), one moves according to a random Markov chain in such way that the invariant measure is the Gibbs measure itself. This first paper describes how the process eventually moves to and between the states of extremal energies. In this complex disordered system, this costs to control the behavior of a Markov chain on a very complicated asymptotically infinite subset of its ``most recurrent'' or ``most s…
reviewed by: Arnaud Le Ny
Mathematics Subject Classification ID: 82C41
Mathematics Subject Classification ID: 60K35
zbMATH DE Number: 1964922
zbMATH Keywords: disordered systems
zbMATH Keywords: high complexity
zbMATH Keywords: random energy model
zbMATH Keywords: Glauber dynamics
zbMATH Keywords: metastability
zbMATH Keywords: ageing
MaRDI profile type: Publication
full work available at URL: s00220-003-0798-4
OpenAlex ID: W2011635203
DOI: 10.1007/S00220-003-0798-4
description: scientific article; zbMATH DE number 1964922