On $q$-analogs of some integrals over GUE
Combinatorics
2014-10-13 v2 Mathematical Physics
Classical Analysis and ODEs
math.MP
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 -analog of the Gaussian unitary ensemble, using -Hermite polynomials and coefficient extraction instead of integration. This way we derive -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 vertices by genus.
Keywords
Cite
@article{arxiv.1407.7960,
title = {On $q$-analogs of some integrals over GUE},
author = {Praveen S. Venkataramana},
journal= {arXiv preprint arXiv:1407.7960},
year = {2014}
}