Roger Van Peski, Columbia University

Title: Random matrices, random partitions, and random groups

Abstract: In probability and mathematical physics we seek to find new phenomena, like Gaussian behavior in the Central Limit Theorem, which occur ubiquitously—in this case, for sums of independent random variables, independent of finer details of what their distributions are. Often these phenomena are first observed in physics or number theory, and the quest to understand them probabilistically can be roughly divided into two steps:

(1) Probing the limit by computations with the nicest possible probabilistic models, where extra algebraic structure pre-limit makes taking the limit possible, and
(2) Developing robust techniques to prove that the limit seen in nice cases indeed occurs universally.

Often (1) is not simply superseded by (2), but instead nontrivial interplay develops between the algebraic structures and the robust analytic techniques.

I will discuss recent and ongoing chapters of this story concerning ‘discrete’ random matrices with entries living in the integers Z, p-adic integers Z_p, or finite fields F_q. Here the players are

(1) Symmetric polynomials and random integer partitions defined using them (Macdonald processes), connecting to random matrices through spherical functions on p-adic groups, and
(2) Asymptotics of random abelian groups, and the ‘moment method’ (and its newer versions) used for proving their universality.

No background apart from undergraduate probability and algebra will be assumed.