Non-differentiability of the effective potential and the replica symmetry breaking in the random energy model

TL;DR

This paper derives the exact effective potential for the two-replica system of the random energy model, revealing non-analytic behavior associated with replica symmetry breaking and glass phase condensation.

Contribution

It provides an exact derivation of the effective potential and links non-analyticity to replica symmetry breaking in the random energy model.

Findings

Effective potential is analytic at high temperature.

Non-analyticity appears at low temperature when replicas coincide.

Non-analyticity indicates replica symmetry breaking and glass condensation.

Abstract

The effective potential for the two-replica system of the random energy model is exactly derived. It is an analytic function of the magnetizations of two replicas, $\varphi^1$ and $\varphi^2$ in the high-temperature phase. In the low-temperature phase, where the replica symmetry breaking takes place, the effective potential becomes non-analytic when $\varphi^1=\varphi^2$. The non-analyticity is considered as a consequence of the condensation of the Boltzmann measure, which is a typical property of a glass phase.