Limit Theorems for the Alloy-type Random Energy Model

TL;DR

This paper investigates the asymptotic behavior of the partition function in a generalized random energy model with mixed normal distributions, revealing phase transitions and stable law convergence across different parameter domains.

Contribution

It introduces a comprehensive phase diagram for the alloy-type REM, detailing the limit laws and phase transitions based on the mixture distribution parameters.

Findings

Partition function converges to stable distributions in various domains.

Phase transitions occur at critical surfaces between different stable laws.

The phase diagram characterizes the model's limit behavior across parameter space.

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 \(\sigma \ne 1\). The phase space \((a,\sigma) \subset \R \times \R_+\) 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 \(\alpha-\)stable with \(\alpha \in (1,2)\), after to 1-stable, and finally to \(\alpha-\)stable with \(\alpha \in (0,1).\) The corresponding phase diagram is the central result of this paper.