Analysis Student Seminar

Wednesday, July 7, 2021 3:00 pm - 3:00 pm EDT

Daniel Perales Anaya, Department of Pure Mathematics, University of Waterloo

"On the anti-commutator of two free random variables"

Let a, b be freely independent random variables in a non-commutative probability space. Based on some considerations on bipartite graphs, we provide a formula to express the n-th free cumulant of the anticommutator ab+ba as a sum indexed by a subset Y_{2n} of non-crossing partitions of {1,2,...,2n}. The study of the sets Y_{2n}  yields new results regarding the distribution of ab+ba. For instance, the size |Y_{2n}| is closely related to the case when a,b have a Marchenko-Pastur (free Poisson) distribution of parameter 1. The talk is based on the preprint arXiv:2101.09444.

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