Recorded 28 February 2025. Félix Parraud of Queen's University presents "The spectrum of tensor of random and deterministic matrices" at IPAM's Free Entropy Theory and Random Matrices Workshop. Abstract: In this talk, we consider operator-valued polynomials in Gaussian Unitary Ensemble random matrices. I will explain a new strategy to bound its Lp -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(N2/3) . As a consequence, if the coefficients are M -dimensional matrices with M=exp(o(N2/3)) , 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. Learn more online at: https://www.ipam.ucla.edu/programs/workshops/free-entropy-theory-and-random-matrices/