  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. The main purpose of this paper is to 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 $n \le\exp (N^\alpha)$ for some explicit $\alpha >0$.
Such results had been obtained by very different techniques for various
regimes, all falling in the category $n\ll N$. 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. Firstly, it applies for arbitrary polynomials and not
only linear polynomials, and secondly, it applies for coefficients of an
arbitrary $C^*$-algebra with non-negative joint moments and not only for
non-negative real numbers.
