Eigenvalues of random matrices with generalised correlations: a path integral approach

TL;DR

This paper develops a path integral method to analytically determine the eigenvalue spectrum of large random matrices with general correlations, revealing how non-diagonal correlations influence system stability.

Contribution

It introduces a novel path integral approach to analyze eigenvalues of correlated random matrices, providing a closed-form expression for the leading eigenvalue.

Findings

Correlations beyond diagonal elements significantly affect eigenvalues.

The method yields a simple formula for the largest eigenvalue.

Non-trivial correlations can alter system stability predictions.

Abstract

Random matrix theory allows one to deduce the eigenvalue spectrum of a large matrix given only statistical information about its elements. Such results provide insight into what factors contribute to the stability of complex dynamical systems. In this letter, we study the eigenvalue spectrum of an ensemble of random matrices with correlations between any pair of elements. To this end, we introduce an analytical method that maps the resolvent of the random matrix onto the response functions of a linear dynamical system. The response functions are then evaluated using a path integral formalism, enabling us to make deductions about the eigenvalue spectrum. Our central result is a simple, closed-form expression for the leading eigenvalue of a large random matrix with generalised correlations. This formula demonstrates that correlations between matrix elements that are not diagonally opposite, which are often neglected, can have a significant impact on stability.