From higher order free cumulants to non-separable hypermaps

TL;DR

This paper simplifies the functional relations between higher order free cumulants and moments, revealing their combinatorial interpretation through generating functions of planar non-separable hypermaps with specific vertex and edge configurations.

Contribution

It provides a simplified derivation of the relations between free cumulants and moments and connects them to the enumeration of non-separable hypermaps, including explicit results for third order cumulants.

Findings

Simplified the functional relations between free cumulants and moments.

Connected these relations to generating functions of non-separable hypermaps.

Explicitly derived the case of third order free cumulants.

Abstract

Higher order free moments and cumulants, introduced by Collins, Mingo, \'Sniady and Speicher in 2006, describe the fluctuations of unitarily invariant random matrices in the limit of infinite size. The functional relations between their generating functions were only found last year by Borot, Garcia-Failde, Charbonnier, Leid and Shadrin and a combinatorial derivation is still missing. We simplify these relations and show how their combinatorial derivation reduces to the computation of generating functions of planar non-separable hypermaps with prescribed vertex valencies and weighted hyper-edges. The functional relations obtained by Borot et al. involve some remarkable simplifications, which can be formulated as identities satisfied by these generating functions. The case of third order free cumulants, whose combinatorial understanding was already out of reach, is derived explicitly.