Crossover behaviour of a one dimensional Random Energy Model

TL;DR

This paper introduces a one-dimensional generalization of the Random Energy Model with spatial correlations, analyzing its crossover from 1D to mean-field behavior and discovering different phase transition characteristics.

Contribution

It formulates a finite-dimensional REM with geometry and correlations, exploring the crossover behavior and phase transitions in one dimension.

Findings

Mean field limit reproduces original REM behavior

Second model exhibits a first order phase transition

Identification of crossover from 1D to mean-field behavior

Abstract

In this note we formulate a finite dimensional generalization of the Random Energy Model (REM) where we introduce a geometry and spatial correlations between energies. We study the model in dimension one by transfer matrix techniques and we look at the crossover from one dimensional to mean-field behaviour. In a first version of the model the mean field limit reproduces the behaviour of the original REM, while a second version of the model exhibits a first order phase transition with a finite latent heat.