I will present a universal transformation that can be realised through monotone Hurwitz numbers. From the point of view of combinatorics, it (fully) simplifies the boundaries of maps and removes one colour from constellations. From the perspective of the topological recursion, it performs a symplectic transformation to the spectral curve (x,y). Finally, in probability, it turns the moments into free cumulants. Expressing the transformation as the action of an operator in the Fock space allows us to find functional relations between higher order moments and free cumulants, which solves an open problem formulated by Collins, Mingo, Sniady et Speicher when they were developing the theory of second order freeness which generalises the R transform of Voiculescu. The notion of higher genus objects appears naturally in the context of combinatorics and topological recursion, which leads us to introduce surfaced free cumulants that capture all order asymptotic expansions of random matrices in presence of unitary invariance.

Based on joint work with Gaëtan Borot, Severin Charbonnier, Felix Leid and Sergey Shadrin: 2112.12184 and ongoing