Exact Analytic Continuation with Respect to the Replica Number in the Discrete Random Energy Model of Finite System Size

TL;DR

This paper derives an exact expression for the moments of the partition function in a finite-size discrete random energy model, revealing a phase transition in replica number space that clarifies replica symmetry breaking without the replica trick.

Contribution

It provides a novel exact analytic expression for the partition function moments valid for finite systems and complex replica numbers, enabling direct analysis of replica symmetry breaking.

Findings

Phase transition at a specific real replica number at low temperatures

Analytic expression validated through numerical confirmation

Clarifies replica symmetry breaking scenario without the replica trick

Abstract

An expression for the moment of partition function valid for any finite system size $N$ and complex power $n (\Re(n)>0)$ is obtained for a simple spin glass model termed the {\em discrete random energy model} (DREM). We investigate the behavior of the moment in the thermodynamic limit $N \to \infty$ using this expression, and find that a phase transition occurs at a certain real replica number when the temperature is sufficiently low, directly clarifying the scenario of replica symmetry breaking of DREM in the replica number space {\em without using the replica trick}. The validity of the expression is numerically confirmed.