Universal relation between Green's functions in random matrix theory

TL;DR

This paper establishes a universal relation between the one-point and connected two-point Green's functions in random matrix theory, independent of the probability distribution for a broad class of matrices.

Contribution

It proves a new universal relation linking Green's functions in random matrix theory, extending previous universality results to a broader class of distributions and Hamiltonian models.

Findings

The relation is universal across various distributions.

It applies to Hamiltonians with deterministic and random parts.

The relation involves a second derivative of a logarithmic expression.

Abstract

We prove that in random matrix theory there exists a universal relation between the one-point Green's function $G$ and the connected two- point Green's function $G_c$ given by \vfill $ N^2 G_c(z,w) = {\part^2 \over \part z \part w} \log (({G(z)- G(w) \over z -w}) + {\rm {irrelevant \ factorized \ terms.}} $ This relation is universal in the sense that it does not depend on the probability distribution of the random matrices for a broad class of distributions, even though $G$ is known to depend on the probability distribution in detail. The universality discussed here represents a different statement than the universality we discovered a couple of years ago, which states that $a^2 G_c(az, aw)$ is independent of the probability distribution, where $a$ denotes the width of the spectrum and depends sensitively on the probability distribution. It is shown that the universality proved here also holds for the more general problem of a Hamiltonian consisting of the sum of a deterministic term and a random term analyzed perturbatively by Br\'ezin, Hikami, and Zee.