P-2021-07
GAËTAN BOROT, SÉVERIN CHARBONNIER, ELBA GARCIA-FAILDE, FELIX LEID, AND SERGEY SHADRIN
ANALYTIC THEORY OF HIGHER ORDER FREE CUMULANTS.
Abstract: We establish the functional relations between generating series of higher order free cumu-
lants and moments in higher order free probability, solving an open problem posed fteen years ago
by Collins, Mingo, Śniady and Speicher. We propose an extension of free probability theory, which
governs the all-order topological expansion in unitarily invariant matrix ensembles, with a corre-
sponding notion of free cumulants and give as well their relation to moments via functional relations.
Our approach is based on the study of a master transformation involving double monotone Hurwitz
numbers via semi-innite wedge techniques, building on the recent advances of the last-named author
with Bychkov, Dunin-Barkowski and Kazarian.
Preprint series: Institut für Mathematik, Humboldt-Universität zu Berlin (ISSN 0863-0976), 2021-07
36 pp.