Incremental universality of Wigner random matrices

TL;DR

This paper explores the universality properties of Wigner random matrices, revealing that universality has multiple degrees corresponding to the moments of matrix entries, and compares different ensembles in the large-size limit.

Contribution

It introduces a nuanced view of universality in Wigner matrices, linking it to the moments of matrix entries and comparing different ensembles in the large-$n$ limit.

Findings

Universality has multiple degrees, counted by moments of entries.

Universality degrees match across different ensembles.

Comparison of large-$n$ limits of connected correlators.

Abstract

Properties of universality have essential relevance for the theory of random matrices usually called the Wigner ensemble. The issue was analysed up to recent years with detailed and relevant results. We present a slightly different view and compare the large-$n$ limit of connected correlators of distinct ensembles: universality has steps or degrees, precisely counted by the number of probability moments of the matrix entries, which match among distinct ensembles.