In this talk I will present one transformation that appears in very different contexts: combinatorial maps that get simplified, constellations that lose colours, cumulants that get freed, and x and y that get symplectically exchanged in topological recursion. I will explain how to realise all these dualities through a transformation that involves monotone Hurwitz numbers that we call master relation. Expressing the transformation as the action of an operator on the Fock space allows us to find functional relations that relate the generating series of higher order free cumulants and moments, which solves an open problem in free probability and generalises the R transform machinery of Voiculescu. This leads us to introduce a notion of surfaced free cumulants that captures the all-order asymptotic expansion in 1/N of random ensembles of matrices of size N in presence of some unitary invariance. We illustrate our formulas by computing the first few decaying terms of the correlation functions of an ensemble of spiked GUE matrices, going beyond the law of large numbers and the central limit theorem.

This is based on joint work with Gaëtan Borot, Severin Charbonnier, Felix Leid and Sergey Shadrin, and some ongoing work.