Seminar by Alexandru Nica

 Please Note: This seminar will be given in-person.

A theorem with CLT flavour, in the framework of the infinite symmetric group

In a 1995 paper, Philippe Biane proved a limit theorem concerning the sequence of star-generators (1,2), (1,3),...,(1,n),... of the infinite symmetric group.  Biane's result looks like an algebraic version of the Central Limit Theorem (CLT), with Wigner's semicircle law playing the role of limit. In my talk I will start by surveying the mechanism of how such CLT-style results (including CLT itself and its counterpart in free probability) come up in connection to the notion of exchangeability for a sequence of non-commutative random variables.

Returning to star-generators, I will then present such a result obtained in joint work with Jacob Campbell and Claus Koestler (arXiv:2203.01763), which also takes in its input a dimension size d, and outputs a limit law related to the eigenvalue distribution of a random d-times-d Hermitian matrix in the so-called GUE model.  Making d go to infinity in this result retrieves the one of Biane, due to the fact (fundamental in random matrix theory) that the eigenvalue distributions of GUE matrices approach asymptotically the semicircle law of Wigner.