Supersymmetry Approach to Almost Diagonal Random Matrices

TL;DR

This paper introduces a supersymmetric field theory framework for analyzing almost diagonal Gaussian Hermitian random matrices with small off-diagonal elements, enabling calculation of spectral and eigenfunction correlations.

Contribution

It develops a regular virial expansion approach for correlation functions in almost diagonal random matrices, explicitly deriving interaction terms for multiple supermatrices.

Findings

Derived a controlled perturbation theory based on a small parameter B.

Explicitly calculated virial coefficients for 2- and 3-matrix interactions.

Provided a method to compute spectral and eigenfunction correlations in these matrices.

Abstract

We develop a supersymmetric field theoretical description of the Gaussian ensemble of the almost diagonal Hermitian Random Matrices. The matrices have independent random entries H_{ij} with parametrically small off-diagonal elements H_{ij}/H_{ii} ~ B << 1. We derive a regular virial expansion of correlation functions in the number of ``interacting'' supermatrices associated with different sites in the real space and demonstrate that the perturbation theory constructed in this way is controlled by a small parameter B. General form of the integral expression for the m-th virial coefficient governed by the ``interaction'' of m supermatrices is presented and calculated explicitly in the cases of 2- and 3-matrix ``interaction''. The suggested technique allows us to calculate both the spectral correlations and the correlations of the eigenfunctions taken at different energies and in different space points.