Self-averaging in random systems - liability or asset?

TL;DR

This paper investigates the conditions under which self-averaging in quenched random systems holds or breaks down, proposing that signal-to-noise ratios can identify phase boundaries such as the ferromagnetic transition.

Contribution

It introduces a method to use signal-to-noise ratios of bound quantities to detect phase boundaries in random systems, especially when correlation lengths are comparable to system size.

Findings

Certain bound quantities are not self-averaging near phase boundaries.

Signal-to-noise ratios can effectively identify the ferromagnetic phase boundary.

The approach compares favorably with traditional phase boundary measures.

Abstract

The study of quenched random systems is facilitated by the idea that the ensemble averages describe the thermal averages for any specific realization of the couplings, provided the system is large enough. Careful examination suggests that this idea might have a follow, when the correlation length becomes of the order of the size of the system. We find certain bound quantities are not self-averaging when the correlation length becomes of the order of the size of the system. This suggests that the strength of self-averaging, expressed in terms of properly chosen signal to noise ratios, may serve to identify phase boundaries. This is demonstrated by using such signal to noise ratios to identify the boundary of the ferromagnetic phase and compare the findings with more traditional measures.