On the operator norm of non-commutative polynomials in deterministic matrices and iid GUE matrices

Abstract

Let X N = (X N 1 , . . . , X N d ) be a d-tuple of N × N independent GUE random matrices and Z NM be any family of deterministic matrices in M N (C) ⊗ M M (C). Let P be a self-adjoint non-commutative polynomial. A seminal work of Voiculescu shows that the empirical measure of the eigenvalues of P (X N ) converges towards a deterministic measure defined thanks to free probability theory. Let now f be a smooth function, the main technical result of this paper is a precise bound of the difference between the expectation ofand its limit when N goes to infinity. If f is six times differentiable, we show that it is bounded by M 2 f C 6 N −2 . As a corollary, we obtain a new proof and slightly improve a result of Haagerup and Thorbjørnsen, later developed by Male, which gives sufficient conditions for the operator norm of a polynomial evaluated in (X N , Z NM , Z NM * ) to converge almost surely towards its free limit.

Strong convergence of tensor products of independent G.U.E. matrices

“…Recent works established results of this nature, dealing with matrices X M , where the dimension of the G.U.E. matrices Y i 's is M and M = O(N 1/4 ) in [14], M = O(N 1/3 ) in [5], and M = o(N/(log N) 3 ) in [2]. As mentioned for instance in [2], this does not suffice for the purpose of [8], which requires M = N (see [2,Proposition 9.3]).…”

Asymptotic Expansion of Smooth Functions in Polynomials in Deterministic Matrices and iid GUE Matrices

“…Until recently, this approach was rarely considered due to the difficulty that working with non-analytic functions bring, although there exist some previous results, see [21] and [27]. More recently though, in [18] we introduced a new approach which consist in interpolating our random matrices with free operators. This approach was refined in [36] where we proved an asymptotic expansion for polynomials in independent GUE matrices and deterministic matrices.…”

“…This approach was refined in [36] where we proved an asymptotic expansion for polynomials in independent GUE matrices and deterministic matrices. In [37], we used the heuristic of [18] to study polynomials of independent Haar unitary matrices and deterministic matrices. Thus by combining the different tools used in those papers, we prove an asymptotic expansion in the unitary case.…”

“…Since there exist no such formula for the eigenvalues of a polynomial in independent Haar unitary matrices, we cannot adapt this proof. Instead we rely on the strategy developed in [18,37,36]. The main idea is to interpolate Haar unitary matrices and free Haar unitaries with the help of a free unitary Brownian motion.…”

“…By doing so, they introduced the notion of strong convergence (see Definition 2.1). For a detailed history of this type of results, we refer to the introduction of [18]. In 2012, Collins and Male used those results to prove that the spectrum of P (U N , U N * , Z N ) converges for the Hausdorff distance towards an explicit subset of R. We summarized this result in Equation (2).…”