Replica Fourier Transform: Properties and Applications

TL;DR

The paper introduces the Replica Fourier Transform, a mathematical tool for analyzing ultrametric structures in disordered systems, and demonstrates its application in diagonalizing Hessian matrices in spin glass models.

Contribution

It provides a systematic definition of the Replica Fourier Transform and illustrates its use in analyzing Gaussian fluctuations in spin glass theories.

Findings

Diagonalization of Hessian matrices in Replica Fourier Space

Analytical computation of Hessian for spherical spin glass models

Application of the transform to disordered systems analysis

Abstract

The Replica Fourier Transform is the generalization of the discrete Fourier Transform to quantities defined on an ultrametric tree. It finds use in con- junction of the replica method used to study thermodynamics properties of disordered systems such as spin glasses. Its definition is presented in a system- atic and simple form and its use illustrated with some representative examples. In particular we give a detailed discussion of the diagonalization in the Replica Fourier Space of the Hessian matrix of the Gaussian fluctuations about the mean field saddle point of spin glass theory. The general results are finally discussed for a generic spherical spin glass model, where the Hessian can be computed analytically.