Progress in the Combinatorial Understanding of Higher-Order Free Cumulants

Explore the latest advancements in the combinatorial understanding of higher-order free cumulants in this 48-minute lecture presented at the Erwin Schrödinger International Institute for Mathematics and Physics (ESI). Delve into the world of unitarily invariant random matrices and their fluctuations in the infinite size limit, as described by higher-order free moments and cumulants. Trace the evolution of this field from the 2006 introduction by Collins, Mingo, Śniady, and Speicher to the 2021 functional relations derived through Fock space computations. Discover the speaker's recent progress in generalizing the combinatorial derivation for higher orders, with a focus on the crucial decomposition of bipartite planar maps into non-separable hypermaps. Gain insights into this complex mathematical topic, which bridges non-commutative geometry and topological recursion, as part of the ESI's specialized workshop series.