Title: Distribution of powers of random unitary matrices and hyperplane arrangements 


Abstract: Let X be a n by n unitary matrix, drawn at random according to the Haar measure on U_n, and let m be a natural number. What can be said about the distribution of X^m and its eigenvalues? 
The density of the distribution \tau_m of X^m can be written as a linear combination of irreducible characters of U_n, where the coefficients are the Fourier coefficients of \tau_m. Classical results in random matrix theory, such as the seminal work of Diaconis--Shahshahani, often focus on the low-dimensional Fourier coefficients of \tau_m. In this talk, I will focus on high-dimensional spectral information about \tau_m. For example: 
(a) Can one give sharp estimates on the rate of decay of its Fourier coefficients? 
(b) For which values of p, is the density of \tau_m  L^p-integrable? 
Building on the work of Rains about the distribution of X^m, we will see how Item (a) is equivalent to a branching problem in the representation theory of certain compact homogeneous spaces, and how (b) is equivalent to a geometric problem about the singularities of certain varieties called (Weyl) hyperplane arrangements. I will answer these questions and further discuss the geometric approach. 
Based on joint works with Julia Gordon and Yotam Hendel and with Nir Avni and Michael Larsen (see [GGH] and [AGL]).


Livestream/Recording Link: https://huji.cloud.panopto.eu/Panopto/Pages/Viewer.aspx?id=e176f953-c1b8-4ef0-bfac-b2e2005cb714