On analyticity with respect to the replica number in random energy models I: an exact expression of the moment of the partition function

TL;DR

This paper derives an exact formula for the moments of the partition function in finite-size random energy models, enabling analysis of analyticity breaking related to replica symmetry breaking in statistical physics.

Contribution

It provides a new exact expression for the moments of the partition function in finite systems, extending previous work and facilitating analysis of analyticity and replica symmetry breaking.

Findings

Exact expression valid for finite systems

Analyticity breaking linked to one-step replica symmetry breaking

Numerical confirmation of the formula's validity

Abstract

We provide an exact expression of the moment of the partition function for random energy models of finite system size, generalizing an earlier expression for a grand canonical version of the discrete random energy model presented by the authors in Prog. Theor. Phys. 111, 661 (2004). The expression can be handled both analytically and numerically, which is useful for examining how the analyticity of the moment with respect to the replica numbers, which play the role of powers of the moment, can be broken in the thermodynamic limit. A comparison with a replica method analysis indicates that the analyticity breaking can be regarded as the origin of the one-step replica symmetry breaking. The validity of the expression is also confirmed by numerical methods for finite systems.