Progress in the combinatorial understanding of higher-order free cumulants

Luca Lionni (U Heidelberg)

Apr 28. 2023, 10:30 — 11:15

Higher order free moments and cumulants describe the fluctuations of unitarily invariant random matrices in the limit of infinite size. They have been introduced at all orders in 2006 by Collins, Mingo, Śniady and Speicher (CMSS), who also derived combinatorially the corresponding functional relations for the second order. On the other hand, the functional relations for the higher orders have been obtained in 2021 by Borot, Garcia-Failde, Charbonnier, Leid and Shadrin by Fock space computations, and a combinatorial derivation is still missing. I will present some recent progress in generalizing the combinatorial derivation of CMSS for higher orders. The key step is a decomposition of bipartite planar maps - certain graphs embedded on the 2-sphere - in non-separable hypermaps.

Further Information
ESI Boltzmann Lecture Hall
Associated Event:
Non-commutative Geometry meets Topological Recursion (Workshop)
Gaëtan Borot (HU Berlin)
Elba Garcia Failde (Sorbonne U, Paris)
Harald Grosse (U of Vienna)
Masoud Khalkhali (Western U, Ontario)
Hannah Markwig (U Tübingen)
Raimar Wulkenhaar (U Münster)