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.

We identify the Kontsevich-Penner matrix integral, for finite size n, with the isomonodromic tau function of a 3 × 3 rational connection on the Riemann sphere with n Fuchsian singularities placed in correspondence with the eigenvalues of the external field of the matrix integral. By formulating the isomonodromic system in terms of an appropriate Riemann-Hilbert boundary value problem, we can pass to the limit n → ∞ (at a formal level) and identify an isomonodromic system in terms of the Miwa variables, which play the role of times of a KP hierarchy. This allows to derive the String and Dilaton equations via a purely Riemann-Hilbert approach.The expression of the formal limit of the partition function as an isomonodromic tau function allows us to derive explicit closed formulae for the correlators of this matrix model in terms of the solution of the Riemann Hilbert problem with all times set to zero. These correlators have been conjectured to describe the intersection numbers for Riemann surfaces with boundaries, or open intersection numbers.Lemma 2.22. The jump matrix J(λ) (2.44) admits formal meromorphic extension in D + . More precisely

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The Brezin-Gross-Witten tau function is a tau function of the KdV hierarchy which arises in the weak coupling phase of the Brezin-Gross-Witten model. It falls within the family of generalized Kontsevich matrix integrals, and its algebro-geometric interpretation has been unveiled in recent works of Norbury. This tau function admits a natural extension, called generalized Brezin-Gross-Witten tau function. We prove that the latter is the isomonodromic tau function of a 2 × 2 isomonodromic system and consequently present a study of this tau function purely by means of this isomonodromic interpretation. Within this approach we derive effective formulae for the generating functions of the correlators in terms of simple generating series, the Virasoro constraints, and discuss the relation with the Painlevé XXXIV hierarchy.