Laguerre Ensemble: Correlators, 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.

arXiv comment41 pages. v5: the value of the normalization constant $C_N$ in Theorem 1.5 has been corrected. No other result in the paper is affected by this change

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Cited by in corpus (8)

• Matrix Kesten Recursion, Inverse-Wishart Ensemble and Fermions in a Morse Potential
• Jacobi Ensemble, Hurwitz Numbers and Wilson Polynomials
• The classical $\beta$-ensembles with $\beta$ proportional to $1/N$: from loop equations to Dyson's disordered chain
• Moments of Generalized Cauchy Random Matrices and continuous-Hahn Polynomials
• Grothendieck's Dessins d'Enfants in a Web of Dualities. III
• Tau-functions for the Ablowitz--Ladik hierarchy: the matrix-resolvent method
• Symmetric Function Theory and Unitary Invariant Ensembles
• $\beta$-ensembles and higher genera Catalan numbers