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Given two polynomials $p(x), q(x)$ of degree $d$, we give a combinatorialformula for the finite free cumulants of $p(x)\boxtimes_d q(x)$. We show thatthis formula admits a topological expansion in terms of non-crossingmulti-annular permutations on surfaces of different genera.  This topological expansion, on the one hand, deepens the connection betweenthe theories of finite free probability and free probability, and in particularproves that $\boxtimes_d$ converges to $\boxtimes$ as $d$ goes to infinity. Onthe other hand, borrowing tools from the theory of second order freeness, weuse our expansion to study the infinitesimal distribution of certain familiesof polynomials which include Hermite and Laguerre, and draw some connectionswith the theory of infinitesimal distributions for real random matrices.  Finally, building off our results we give a new short and conceptual proof ofa recent result [Steinerberger (2020), Hoskins and Kabluchko (2020)] thatconnects root distributions of polynomial derivatives with free fractionalconvolution powers.

Finite Free Cumulants: Multiplicative Convolutions, Genus Expansion and Infinitesimal Distributions