Andrew Campbell
IST Austria
Title
Finite free probability and symmetric functions in random variables
Abstract
We will discuss limits of elementary symmetric polynomial evaluated in random variables. We will particularly focus on this problem in the context of finite free probability and in the case when the entries are independent. If the entries have mean 0 and variance 1, then it is known that limit can be expressed in terms of Hermite polynomials in standard Gaussian random variables. We will see that finite free probability provides an efficient way to extend this result to random variables in the domains of attraction of more general infinitely divisible distributions. In this regime the Hermite polynomials are replaced with a random Appell sequence with distribution determined by the infinitely divisible distribution. Time permitting we will discuss other symmetric functions, connections to random matrix theory, and moving beyond independent random variables. Based on joint work with Octavio Arizmendi and Katsunori Fujie.