Laguerre Ensemble: Correlators, Hurwitz Numbers and Hodge Integrals

We consider the Laguerre partition function, and derive explicit generating func-tions for connected correlators with arbitrary integer powers oftraces in terms of products ofHahn polynomials. It was recently proven in [22] that correlators have a topological expansionin terms of weakly or strictly monotone Hurwitz numbers, that can be explicitly computed fromour formulæ. As a second result we identify the Laguerre partition function with only positivecouplings and a special value of the parameterα=−1/2 with the modified GUE partitionfunction, which has recently been introduced in [28] as a generating function for Hodge inte-grals. This identification provides a direct and new link between monotone Hurwitz numbersand Hodge integrals.