Role of the Finite Replica Analysis in the Mean-Field Theory of Spin Glasses

TL;DR

This thesis explores the replica method's role in the mean-field theory of spin glasses, focusing on replica symmetry breaking, its mathematical and physical aspects, and its application to various models including the Ising perceptron.

Contribution

It provides a comprehensive review of the replica method, emphasizing the significance of replica symmetry breaking and its implications for spin-glass models and related systems.

Findings

Detailed analysis of replica symmetry breaking

Application to spin-glass models and Ising perceptron

Insights into mathematical and physical descriptions

Abstract

In this thesis, we review and examine the replica method from several viewpoints. The replica method is a mathematical technique to calculate general moments of stochastic variables. This method provides a systematic way to evaluate physical quantities and becomes one of the most important tools in the theory of spin glasses and in the related discipline including information processing tasks. In spite of the effectiveness of the replica method, it is known that several problems exist in the procedures of the method itself. The replica symmetry breaking is the central topic of those problems and is the main issue of this thesis. To elucidate this point, we review the recent progress about the replica symmetry breaking including its physical and mathematical descriptions in detail. Based on those descriptions, several spin-glass models and Ising perceptron are deeply investigated.