Sampling unitary invariant ensembles

TL;DR

This paper introduces a Monte Carlo sampling algorithm for unitary invariant random matrix ensembles, enabling the computation of complex statistics and revealing unexpected phenomena through simulations.

Contribution

It presents a novel algorithm leveraging orthogonal polynomial kernels to efficiently sample eigenvalues of unitary invariant ensembles.

Findings

Enables calculation of statistics beyond analytical methods

Reveals unexpected phenomena in eigenvalue simulations

Provides a practical tool for studying random matrix ensembles

Abstract

We develop an algorithm for sampling from the unitary invariant random matrix ensembles. The algorithm is based on the representation of their eigenvalues as a determinantal point process whose kernel is given in terms of orthogonal polynomials. Using this algorithm, statistics beyond those known through analysis are calculable through Monte Carlo simulation. Unexpected phenomena are observed in the simulations.