On the existence of a glass transition in a Random Energy Model

TL;DR

This paper investigates a generalized Random Energy Model, demonstrating that a glass transition typically exists with smooth local energy distributions but may vanish with discrete distributions as the system size grows, highlighting the role of energy distribution shape.

Contribution

It extends the Random Energy Model by analyzing how the distribution of local energies affects the existence of a glass transition, revealing conditions under which the transition disappears.

Findings

Glass transition exists with smooth energy distributions.

Transition may vanish with discrete energy levels as system size increases.

Statistical independence alone does not guarantee a glass transition.

Abstract

We consider a generalized version of the Random Energy Model in which the energy of each configuration is given by the sum of $N$ independent contributions ("local energies") with finite variances but otherwise arbitrary statistics. Using the large deviation formalism, we find that the glass transition generically exists when local energies have a smooth distribution. In contrast, if the distribution of the local energies has a {Dirac mass} at the minimal energy (e.g., if local energies take discrete values), the glass transition ceases to exist if the number of energy levels grows sufficiently fast with system size. This shows that statistical independence of energy levels does not imply the existence of a glass transition.