Freezing and extreme value statistics in a Random Energy Model with logarithmically correlated potential

TL;DR

This paper explores the freezing scenario in a circular logarithmically correlated random energy model, deriving new extreme-value statistics for strongly correlated variables and connecting them to Dyson Coulomb gas integrals.

Contribution

It introduces a circular variant of the REM with logarithmic correlations and derives the full distribution of the minimal potential value using the freezing scenario.

Findings

The moments of the partition function relate to Dyson Coulomb gas integrals.

The freezing scenario enables extraction of free energy distributions in both phases.

The minimal potential distribution forms a new class of extreme-value statistics.

Abstract

We investigate some implications of the freezing scenario proposed by Carpentier and Le Doussal (CLD) for a random energy model (REM) with logarithmically correlated random potential. We introduce a particular (circular) variant of the model, and show that the integer moments of the partition function in the high-temperature phase are given by the well-known Dyson Coulomb gas integrals. The CLD freezing scenario allows one to use those moments for extracting the distribution of the free energy in both high- and low-temperature phases. In particular, it yields the full distribution of the minimal value in the potential sequence. This provides an explicit new class of extreme-value statistics for strongly correlated variables, manifestly different from the standard Gumbel class. -