Norm of matrix-valued polynomials in random unitaries and permutations

TL;DR

This paper investigates the operator norm of non-commutative polynomials in large random matrices from groups like the unitary, orthogonal, and symmetric groups, showing they approximate free group algebra norms and establishing universal bounds.

Contribution

It provides a new proof of the Peterson-Thom conjecture and establishes universal lower bounds for operator norms in random matrix models with coefficients in any C*-algebra.

Findings

Operator norms of polynomials in random unitaries converge to free group algebra norms as N grows.

Universal lower bounds for operator norms in random matrices with coefficients in arbitrary C*-algebras.

Extension of Alon-Boppana bounds to non-linear polynomials and broader coefficient classes.

Abstract

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.