# Student Probability Seminar

#### 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 afree 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.