in this talk, we consider operator-valued polynomials in gaussian unitary ensemble random matrices. i will explain a new strategy to bound its $l^p$-norm, up to an asymptotically small error, by the operator norm of the same polynomial evaluated in free semicircular variables as long as $p=o(n^{2/3})$. as a consequence, if the coefficients are $m$-dimensional matrices with $m=exp(o(n^{2/3}))$, then the operator norm of this polynomial converges towards the one of its free counterpart. in particular this provides another proof of the peterson-thom conjecture thanks to the result of ben hayes. the approach that we take in this paper is based on an asymptotic expansion obtained in a previous paper combined with a new result of independent interest on the norm of the composition of the multiplication operator and a permutation operator acting on a tensor of $c^*$-algebras.

by the way, can i assume that the people attending the seminar will be familiar with notions of free probability?