Glauber dynamics of spin glasses at low and high temperature

Abstract

We consider an increasing sequence of finite boxes L ⊂ Z 2 and a reversible stochastic frustrated Ising model having invariant measures satisfying free boundary conditions. We show that the spectral gap associated with the Edwards-Anderson model has a different asymptotic behavior in low and in high temperature. In low temperature, associated with the spectral gap, there is a qualitatively slower relaxation to equilibrium than there is in high temperature. Some geometrical lemmas are employed in the paper to show that some regions are almost independent from their exterior. We use for this aim a Peierls' argument.  2002 Éditions scientifiques et médicales Elsevier SAS AMS classification: 82C44; 82C26; 82C22 RÉSUMÉ.-Nous considérons, dans cet article, une suite strictement croissante de boites finies L ⊂ Z 2 et un modèle d'Ising stochastique réversible non ferromagnétique ayant des mesures invariantes avec conditions de frontière libre. Nous montrons que le frou spectral associé au modèle d'Edwards-Anderson a des comportements asymptotiques différents à haute et basse température. A basse température, la relaxation vers l'équilibre est qualitativement plus lente qu'à haute température. Nous employons dans cet article des lemmes géomêtriques pour montrer que certaines régions sont presque indépendantes de leur exterieur. À cette fin, nous utilisons un argument de type Peierls.  2002 Éditions scientifiques et médicales Elsevier SAS

On the spectral gap of spherical spin glass dynamics

“…For models on the hypercube, there is an exponential lower bound on the spectral gap in terms of an intrinsic quantity [45]. In the short range setting, there are some results from e.g., [35,25]. However, for the classical mean-field models of the p-spin models on {±1} N and S N , the study of the spectral gap of Glauber/Langevin dynamics has remained largely open.…”

Stochastic Ising model with flipping sets of spins and fast decreasing temperature

“…There are important results for the convergence of the system to the Gibbs state in the case of low temperature (see e.g. [6,7,9,20]) or high temperature (see e.g. [6,8,19,21]).…”

“…[6,7,9,20]) or high temperature (see e.g. [6,8,19,21]). In particular, in [6], in the framework of random interactions, it is shown a different approach to the equilibrium measure in relation to the temperature.…”

“…The geometric structure of the graph is also relevant in both the cases of zero-and low constant temperature. In [6,14] some of the main results come out from a combinatorial-geometric interpretation of the underlying graphs.…”

“…[6,8,19,21]). In particular, in [6], in the framework of random interactions, it is shown a different approach to the equilibrium measure in relation to the temperature. We believe that there is a connection between the behaviour of the stochastic Ising model with α ∈ (0, 1) at low, constant and positive temperature and at zero-temperature.…”