On $q$-analogs of some integrals over GUE

TL;DR

This paper introduces a formal $q$-analog of the Gaussian unitary ensemble using $q$-Hermite polynomials, deriving $q$-analogs of classical eigenvalue statistics and connecting to combinatorial map enumeration.

Contribution

It develops a novel $q$-analog framework for GUE, replacing integration with coefficient extraction, and extends classical eigenvalue statistics to this new setting.

Findings

Derived $q$-analogs for eigenvalue statistics.

Connected $q$-analogs to Harer-Zagier formula.

Provided combinatorial interpretations via $q$-analogs.

Abstract

Statistics over the Gaussian unitary ensemble and the Wishart ensemble of random matrices often have nice closed-form expressions. These are related to multivariate extensions of the Hermite, Laguerre, and Jacobi polynomials, which often occur in the study of these ensembles. In the paper, we develop a formal $q$-analog of the Gaussian unitary ensemble, using $q$-Hermite polynomials and coefficient extraction instead of integration. This way we derive $q$-analogs for many well-known eigenvalue statistics. One of these is related to the Harer-Zagier formula, which uses a matrix integral to count the number of unicellular maps on $n$ vertices by genus.