Glauber dynamics of the random energy model. I: Metastable motion on the extreme states
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 stable'' states on appropriate time scales. The REM is thus considered as a complicated random walk on the hypercube between the most profound traps and written as a jump process. According to physicists' predictions it is obvious that this process jumps with the uniform distribution on the extremal states, and that the time between the visit of two extreme points is asymptotically exponentially distributed. This last property is interpreted as an ageing phenomena (infinite states communicate at the same time scale) and is the subject of the second paper. These two papers provide a first mathematical description of these results in a weak asymptotic form enough to prove the emergence of a non-Markovian limit ultimately responsible for ageing in agreement with another paradigm, the so-called Bouchaud's trap model (and in a similar way). The first paper focuses on the weak asymptotic form and adapts metastability techniques in stochastic dynamics of disordered systems. The sequence of visits on the selected set of the states of lowest energy is described asymptotically by a simple discrete time Markov chain on this set which jumps from one point to the next with the uniform distribution in a similar way as in Bouchaud's model. Two more questions are considered: the mean entrance time and entrance law in the extreme set, and mean entrance time between points of this set (which depend only of the starting point). In the low temperature regime, this mean time to reach appears to be proportional to the smallest waiting time of this set. The proof uses an adaptation of recent analysis of metastability in similar Markov chains by one of the authors, with a different and careful treatment of the time scales, where different choices distinguish ageing from metastability: for the ageing phenomenon, infinite states communicate at the same time scale without any gap between favourite and non favourite sites, while long metastable time scales were assumed to be well separated from the shortest time scales on which the process may stay away. According to the authors, the general methodology developed could be of use in many other contexts of the dynamics of complex systems.
scientific article; zbMATH DE number 1964922