Sixty Years of Moments for Random Matrices

TL;DR

This paper reviews the historical and recent developments in the method of moments for analyzing the eigenvalue distributions and operator norms of real symmetric random matrices, highlighting variations for different ensembles.

Contribution

It provides an accessible overview of classical and recent results on the limiting spectral distributions of various random matrix ensembles using the method of moments.

Findings

Classical Wigner results for independent, centered, same-variance entries

Extensions to band matrices with variance depending on entry position

Recent results for matrices with exchangeable or Curie-Weiss distributed entries

Abstract

This is an elementary review, aimed at non-specialists, of results that have been obtained for the limiting distribution of eigenvalues and for the operator norms of real symmetric random matrices via the method of moments. This method goes back to a remarkable argument of Eugen Wigner some sixty years ago which works best for independent matrix entries, as far as symmetry permits, that are all centered and have the same variance. We then discuss variations of this classical result for ensembles for which the variance may depend on the distance of the matrix entry to the diagonal, including in particular the case of band random matrices, and/or for which the required independence of the matrix entries is replaced by some weaker condition. This includes results on ensembles with entries from Curie-Weiss random variables or from sequences of exchangeable random variables that have been obtained quite recently.