On analyticity with respect to the replica number in random energy models II: zeros on the complex plane

TL;DR

This paper investigates the analyticity breaking in random energy models by analyzing the complex zeros of the partition function's moments, revealing phase transition-like behavior and providing both analytical and numerical insights.

Contribution

It introduces a detailed analysis of the zeros of the partition function's moments to characterize analyticity breaking, connecting it to phase transition phenomena.

Findings

Zeros form a locus with an incident angle indicating second-order phase transition

Analytical evaluation of zeros near the transition point

Numerical results agree with analytical predictions for finite systems

Abstract

We characterize the breaking of analyticity with respect to the replica number which occurs in random energy models via the complex zeros of the moment of the partition function. We perturbatively evaluate the zeros in the vicinity of the transition point based on an exact expression of the moment of the partition function utilizing the steepest descent method, and examine an asymptotic form of the locus of the zeros as the system size tends to infinity. The incident angle of this locus indicates that analyticity breaking is analogous to a phase transition of the second order. We also evaluate the number of zeros utilizing the argument principle of complex analysis. The actual number of zeros calculated numerically for systems of finite size agrees fairly well with the analytical results.