Large Random Matrices: Eigenvalue Distribution

TL;DR

This paper introduces a recursive method to compute eigenvalue correlation functions of large random hermitian matrices, demonstrating universality in two-point functions and providing explicit formulas for higher-order correlations.

Contribution

It presents a novel recursive approach for calculating all eigenvalue correlation functions in large random hermitian matrices, including explicit formulas for three and four-point functions.

Findings

Two-point eigenvalue correlation function is universal.

Explicit formulas for three and four-point functions are derived.

Higher order correlations depend on matrix distribution parameters.

Abstract

A recursive method is derived to calculate all eigenvalue correlation functions of a random hermitian matrix in the large size limit, and after smoothing of the short scale oscillations. The property that the two-point function is universal, is recovered and the three and four-point functions are given explicitly. One observes that higher order correlation functions are linear combinations of universal functions with coefficients depending on an increasing number of parameters of the matrix distribution.