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The tensor HCIZ integral and its relation to enumerative geometry and free probability

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Luca Lionni
Radboud University/Heidelberg Universität
Tue, 08/02/2022 - 13:45 - 15:30

In a recent paper with Collins and Gurau, we introduced a generalization of the celebrated Harish-Chandra—Itzykson—Zuber (HCIZ) integral to tensors, which finds applications for instance to random geometry, quantum information, and the two topics I will focus on in this talk: enumerative geometry and free probability.
♦ This integral can be shown to be a generating function for a generalization of Hurwitz numbers involving certain transitive factorizations of multiplets of permutations. For the original HCIZ integral, these numbers are known to count certain connected ramified coverings of the 2-sphere, and the question thus naturally arises of what our generalization enumerates. I will give two answers, one in terms of ramified coverings of a collection of 2-spheres that touch at one common node, the other in terms of coverings of the 3-sphere branched over certain subgraphs of the complete bipartite graph K3,p.

♦ Free probability consists in the study of non-commutative random variables, such as random matrices in the limit of infinite size. Free cumulants are one of the central objects of free probability, and it is known that certain fundamental properties of free cumulants can be recovered from the original HCIZ integral. I will present early results on the way that our generalization of the HCIZ integral allows adapting the tools of free probability to the study of large random tensors.


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