The Perturbative Resolvent Method: spectral densities of random matrix ensembles via perturbation theory

TL;DR

The paper introduces the Perturbative Resolvent Method (PRM), a straightforward perturbative technique for deriving spectral densities of various random matrix ensembles in the large-size limit, simplifying existing derivations.

Contribution

The paper presents the PRM, a new perturbative approach that simplifies the calculation of spectral densities for multiple random matrix ensembles using the cavity method.

Findings

Derived the Wigner Semi-circle Law analytically

Obtained the Marchenko-Pastur Law through the method

Calculated spectral densities for product Wishart matrices

Abstract

We present a simple, perturbative approach for calculating spectral densities for random matrix ensembles in the thermodynamic limit we call the Perturbative Resolvent Method (PRM). The PRM is based on constructing a linear system of equations and calculating how the solutions to these equation change in response to a small perturbation using the zero-temperature cavity method. We illustrate the power of the method by providing simple analytic derivations of the Wigner Semi-circle Law for symmetric matrices, the Marchenko-Pastur Law for Wishart matrices, the spectral density for a product Wishart matrix composed of two square matrices, and the Circle and elliptic laws for real random matrices.