Noncrossing hyperchords and free probability

Abstract: Free probability is a noncommutative probability theory introduced by Voiculescu in the 1980s, motivated by operator algebras and free groups, and useful in random matrix theory. Where classical independence relates to the tensor product of algebras, free independence relates to the free product of algebras. Speicher discovered the combinatorial substrate of the theory: noncrossing partitions. He derived the free cumulant-moment relations from Möbius inversion in the incidence algebra of the lattice of noncrossing partitions, and used it, via two reduction procedures, to model free multiplicative convolution. A crucial ingredient, which has no analogue in the classical setting, is the notion of Kreweras complement of a noncrossing partition. In this talk, after a long introduction to these topics, I will explain some more categorical viewpoints. A first step is an operad of noncrossing partitions. A second step is a decomposition space (2-Segal space) Y of noncrossing hyperchords, whose simplicial structure encodes higher versions of Kreweras complementation. The incidence bialgebra of Y is a direct combinatorial model for free multiplicative convolution. It is related to the previous models by the standard simplicial notion of decalage: the first decalage of Y gives the (two-sided bar construction of the) operad, and the second decalage gives the lattice. These two decalages encode precisely Speicher's two reductions.

This is joint work with Kurusch Ebrahimi-Fard, Loïc Foissy, and Frédéric Patras.

Computer sciencealgebraic topologycategory theorylogic

Audience: learners

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