This is joint work with Benoit Collins (Univ. Kyoto). We consider a non-commutative polynomial in several independent $N$-dimensional random unitary matrices, uniformly distributed over the unitary, orthogonal or symmetric groups, and assume that the coefficients are $n$-dimensional matrices. We study the operator norm of this random non-commutative polynomial. We compare it with its counterpart where the the random unitary matrices are replaced by the unitary generators of the free group von Neumann algebra. Our first result is that these two norms are overwhelmingly close to each other in the large $N$ limit, and this estimate is uniform over all matrix coefficients as long as $n\leq \exp (N^α)$ for some explicit $α>0$. Our result provides a new proof of the Peterson-Thom conjecture. Our second result is a universal quantitative lower bound for the operator norm of polynomials in independent $N$-dimensional random unitary and permutation matrices with coefficients in an arbitrary $C^∗$-algebra. A variant of this result for permutation matrices generalizes the Alon-Boppana lower bound in two directions.
(This is a joint-seminar with Kyoto Operator Algebra Seminar.)