Title:Largest Singular Values of Products of \beta-Ensembles

Abstract:Let $U_1,\ldots,U_T$ be independent Haar distributed unitary matrices and foreach $1 \le i \le T$, let $A_i$ be an $N|times N$ submatrix of$U_i$. In this talk, I will discuss the behavior of the largest singularvalues of $A_1 \cdots A_T$ under the regime where the sizes of the matrices$N$ grow linearly with the number of products $T$. The limitingdistribution is described by a Markov process in time $\lim_{N\to\infty}T/N$ which interpolates between the Airy point process and a deterministicparticle configuration. I will also discuss what is known in the analogouscase where we replace unitary with orthogonal or symplectic matrices.