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On $q$-analogs of some integrals over GUE

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.