03-12【Giulio Ruzza】ZOOM会议“可积系统与几何”联合学术报告


报告题目: Correlators of classical hermitian matrix ensembles: discrete orthogonal polynomials and Hurwitz numbers

报告人:Dr. Giulio Ruzza,UCLouvain

报告时间: 2021年3月12日(周五) 16:30-18:00

Zoom: 351 891 2672

Password: 123456

摘要:The correlators of a random matrix ensemble are the expectations of products of traces of powers of the random matrix. In this talk I will focus on the case of hermitian matrix ensembles; in such case, general formulae for the correlators can be given. A careful study of these formulae in the classical cases (Gaussian, Laguerre, and Jacobi) allows to express them in terms of discrete orthogonal polynomials, generalizing known results for one-point correlators.

I will then consider the large size topological expansion of these correlators; it is a classical result of Bessis, Itzykson and Zuber that the correlators in the Gaussian case enumerate ribbon graphs. We extend this result to the Jacobi case, proving that the correlators in this case enumerate certain triple monotone Hurwitz numbers. By a simple limit, we recover a similar result for the Laguerre Ensemble, previously proved by Cunden, Dahlqvist, O'Connell, and Simm.

Finally, I will explain, building on the Hodge-GUE correspondence by Dubrovin, Liu, Yang, and Zhang, how the Laguerre correlators and corresponding Hurwitz numbers are related to Hodge integrals on the moduli spaces of curves.

This is a joint work with Massimo Gisonni and Tamara Grava.