An Extension of Level-spacing Universality

TL;DR

This paper extends Dyson's level-spacing universality to a broader class of Hamiltonians, showing that the level-spacing distribution remains universal even when a deterministic matrix is added to a Gaussian random matrix.

Contribution

It demonstrates that Dyson's short-distance universality applies to Hamiltonians of the form H=H_0+V, with H_0 deterministic and V Gaussian, establishing the independence of P(s) from H_0.

Findings

Dyson's universality holds for H=H_0+V with H_0 deterministic

The n-point correlation function remains a determinant form

Level-spacing distribution P(s) is independent of H_0

Abstract

Dyson's short-distance universality of the correlation functions implies the universality of P(s), the level-spacing distribution. We first briefly review how this property is understood for unitary invariant ensembles and consider next a Hamiltonian H = H_0+ V , in which H_0 is a given, non-random, N by N matrix, and V is an Hermitian random matrix with a Gaussian probability distribution. n-point correlation function may still be expressed as a determinant of an n by n matrix, whose elements are given by a kernel $K(\lambda,\mu)$ as in the H_0=0 case. From this representation we can show that Dyson's short-distance universality still holds. We then conclude that P(s) is independent of H_0.