Limit Theorems for the Alloy-type Random Energy Model

Abstract

In this paper, we consider limit laws for the model, which is a generalisation of the random energy model (REM) to the case when the energy levels have the mixture distribution. More precisely, the distribution of the energy levels is assumed to be a mixture of two normal distributions, one of which is standard normal, while the second has the mean \(\sqrt{n}a\) with some \(a\in \R,\) and the variance \(σ\ne 1\). The phase space \((a,σ) \subset \R \times \R_+\) is divided onto several domains, where after appropriate normalisation, the partition function converges in law to the stable distribution. These domains are separated by the critical surfaces, corresponding to transitions from the normal distribution to \(α-\)stable with \(α\in (1,2)\), after to 1-stable, and finally to \(α-\)stable with \(α\in (0,1).\) The corresponding phase diagram is the central result of this paper.