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 in Cunden et al. (Ann. Inst. Henri Poincaré D, to appear) 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 $$\alpha =-1/2$$ α = - 1 / 2 with the modified GUE partition function, which has recently been introduced in Dubrovin et al. (Hodge-GUE correspondence and the discrete KdV equation. arXiv:1612.02333) as a generating function for Hodge integrals. This identification provides a direct and new link between monotone Hurwitz numbers and Hodge integrals.

Jacobi Ensemble, Hurwitz Numbers and Wilson Polynomials

“…Therefore the results of the present work about the JUE directly imply analogous results for the LUE; these results are already known from [17,31]. See also Remark 2.9.…”

“…This provides an analogue of the classical result of Bessis, Itzykson and Zuber [14] expressing correlators of the Gaussian Unitary Ensemble as a generating function counting ribbon graphs weighted according to their genus, see also [25]. At the same time, it is more similar in spirit (and actually a generalization, see Remark 1.10) of the analogous result for the Laguerre Unitary Ensemble, whose correlators are expressed in terms of double monotone Hurwitz numbers [17], and (for a specific value of the parameter) in terms of Hodge integrals [20,22,31]; in particular in [31] we provide an ELSV-type formula [24] for weighted double monotone Hurwitz numbers in terms of Hodge integrals.…”

“…[21,27]. They are directly related to the theory of tau functions (formal [21] and isomonodromic [9,31]) and to topological recursion theory [5,6,15,28]. Incidentally, similar formulae also appear for matrix models with external source [7,8,[11][12][13]41].…”

On tau-functions for the KdV hierarchy

“…We also note that the matrix resolvent method and some of the above-mentioned methods extend to new situations (see e.g. [14,15,37,38,39,40], [3], [8,9,11,59] and [16,17,18,51]).…”

“…for some integer p ≥ 0, then u elliptic (t) is a special finite-gap solution [36,32] of the KdV hierarchy. Denote by y 2 = S p (λ) (51) the corresponding spectral curve [36,32], where S p (λ) is a polynomial of λ of degree 1+2p with leading coefficient 1. The first few S p are S 0 = λ, S…”