Maps, Hurwitz numbers and formulas for free probability at all genera | Vidéo | Carmin.tv

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Maps, Hurwitz numbers and formulas for free probability at all genera

By Gaëtan Borot

Appears in collection : Random Geometry / Géométrie aléatoire

I will talk about a transformation involving double monotone Hurwitz numbers, which has several interpretations: transformation from maps to fully simple maps, passing from cumulants to free cumulants in free probability, action of an operator in the Fock space, symplectic exchange in topological recursion. In combination with recent work of Bychkov, Dunin-Barkowski, Kazarian and Shadrin, we deduce functional relations relating the generating series of higher order cumulants and free cumulants. This solves a 15-year old problem posed by Collins, Mingo, Sniady and Speicher (the first order is Voiculescu R-transform). This leads us to a general theory of 'surfaced' freeness, which captures the all order asymptotic expansions in unitary invariant random matrix models, which can be described both from the combinatorial and the analytic perspective. Based on https://arxiv.org/abs/2112.12184 with Séverin Charbonnier, Elba Garcia-Failde, Felix Leid and Sergey Shadrin.

Information about the video

Date of recording 18/01/2022
Date of publication 04/02/2022
Institution CIRM
Licence CC BY NC ND
Language English
Audience Researchers
Director(s) Guillaume Hennenfent
Format MP4

Citation data

DOI 10.24350/CIRM.V.19877703
Cite this video Borot, Gaëtan (18/01/2022). Maps, Hurwitz numbers and formulas for free probability at all genera. CIRM. Audiovisual resource. DOI: 10.24350/CIRM.V.19877703
URL https://dx.doi.org/10.24350/CIRM.V.19877703

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Bibliography

BOROT, Gaëtan, CHARBONNIER, Séverin, GARCIA-FAILDE, Elba, et al. Analytic theory of higher order free cumulants. arXiv preprint arXiv:2112.12184, 2021. - https://arxiv.org/abs/2112.12184

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