Laguerre Ensemble: Correlators, Hurwitz Numbers and Hodge Integrals

TL;DR

This paper derives explicit formulas for Laguerre ensemble correlators, links them to Hurwitz numbers and Hodge integrals, and identifies a special case with the modified GUE partition function, establishing new connections in mathematical physics.

Contribution

It provides explicit generating functions for Laguerre correlators and establishes a novel link between Hurwitz numbers and Hodge integrals through the Laguerre partition function.

Findings

Explicit formulas for Laguerre correlators using Hahn polynomials.

Topological expansion of correlators in terms of Hurwitz numbers.

Identification of a special Laguerre partition function with the modified GUE, connecting Hurwitz numbers and Hodge integrals.

Abstract

We consider the Laguerre partition function, and derive explicit generating functions for connected correlators with arbitrary integer powers of traces in terms of products of Hahn polynomials. It was recently proven that correlators have a topological expansion in terms of weakly or strictly monotone Hurwitz numbers, that can be explicitly computed from our formulae. As a second result we identify the Laguerre partition function with only positive couplings and a special value of the parameter $\alpha=-1/2$ with the modified GUE partition function, which has recently been introduced as a generating function of Hodge integrals. This identification provides a direct and new link between monotone Hurwitz numbers and Hodge integrals.