Universality of Random-Matrix Results for non-Gaussian Ensembles

TL;DR

This paper demonstrates that in large random matrices, the statistical properties of energy levels and S-matrix correlations are universal across different non-Gaussian distributions, extending previous universality results.

Contribution

It proves the universality of energy level and S-matrix correlation functions for non-Gaussian ensembles using supersymmetry, generalizing earlier Gaussian-based results.

Findings

Energy level correlations are independent of the distribution in the large N limit.

Correlation functions of S-matrix elements are universal across ensembles.

Universality applies to ensembles with external parameter dependence.

Abstract

We study random-matrix ensembles with a non-Gaussian probability distribution $P(H) \sim \exp (-N {\rm tr }\, V(H))$ where $N$ is the dimension of the matrix $H$ and $V(H)$ is independent of $N$. Using Efetov's supersymmetry formalism, we show that in the limit $N \rightarrow \infty$ both energy level correlation functions and correlation functions of $S$-matrix elements are independent of $P(H)$ and hence universal on the scale of the local mean level spacing. This statement applies to each of the three generic ensembles (unitary, orthogonal, and symplectic). Universality is also found for correlation functions depending on some external parameter. Our results generalize previous work by Brezin and Zee [Nucl.\ Phys.\ B {\bf 402}, 613 (1993)].