Student Probability Seminar

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Sums of Noncommuting Random Variables and Free Harmonic Analysis

Speaker: Ben McKenna

Location: Warren Weaver Hall 805

Date: Thursday, March 8, 2018, 11 a.m.

Synopsis:

What is the distribution of the sum of two random variables? The independent case is classical and fully understood; the distribution is a convolution, and as such can be computed using harmonic analysis. We will answer this question when the random variables are noncommutative. We'll see how to define a``free convolution,'' and explain how this can be computed with the Stieltjes transform (which is of related interest in its own right as a fundamental link between complex analysis and random matrix theory). Our motivating example throughout will be simple random walk on the free group F_d on d letters, and specifically how recurrence/transience depend on d - the noncommutative analogue of Polya's classical theorem on Z^d.

Notes:

No knowledge of free probability will be assumed.