Via the machinery of Feynman diagrams, integrals over hermitian matrices of size N are generating series for maps with boundaries (= discretized surfaces = dual of ribbon graphs), sorted by Euler characteristic. These generating series can be in turned computed by a recursion on the Euler characteristic (the so-called topological recursion): the initial data for the recursion is a rational curve S : {P(x,y) = 0} presented as a branched cover x : S -> P^1, and the generating series turn into meromorphic forms over S with poles at the zeroes of dx. This count includes rather singular maps, e.g. where some vertices are separating/some faces are glued to themselves.

We introduce the notion of maps with simple boundaries, in which the vertices of the boundary cannot be separating. We show that the generating series for this more restricted class of maps can still be represented by hermitian matrix integrals, and satisfies the topological recursion where the initial data (S,x,y) is replaced by (S,y,x). Our results give a interpretation via combinatorics of maps to the symplectic invariance of the topological recursion, and to Voiculescu R-transform and (higher order) free cumulants in free probability.

We also show that there is a universal relation between generating series of maps with ordinary boundaries and maps with simple boundaries, expressed in terms of monotone double Hurwitz numbers. As application, we deduce in the case of the Gaussian matrix model an ELSV-like formula expressing the 2-orbifold monotone Hurwitz numbers as Hodge integrals over the moduli space of curves. I will also discuss the possibility to obtain (so far unknown) explicit relations between higher order free cumulants and cumulants in free probability at the level of generating series.

Based on joint work Elba Garcia-Failde, as well as ongoing work with Elba and Danilo Lewanski.

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