A Note on the Eigenvalue Density of Random Matrices

TL;DR

This paper establishes the eigenvalue density for certain random matrix ensembles by linking it to a variational principle in statistical mechanics, extending previous theorems to new classes of matrices.

Contribution

It proves a general theorem connecting eigenvalue distributions of random matrices to a variational principle, covering previously uncharacterized ensembles.

Findings

Eigenvalue density characterized by a variational principle.

Extension of known results to new random matrix ensembles.

Connection between eigenvalues and classical charges in statistical mechanics.

Abstract

The distribution of eigenvalues of N times N random matrices in the limit N to infinity is the solution to a variational principle that determines the ground state energy of a confined fluid of classical unit charges. This fact is a consequence of a more general theorem, proven here, in the statistical mechanics of unstable interactions. Our result establishes the eigenvalue density of some ensembles of random matrices which were not covered by previous theorems.