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

Abstract

Let \(U^N = (U_1^N,\dots ,U^N_p)\) be a p-tuple of \(N\times N\) independent Haar unitary matrices and \(Z^{NM}\) be any family of deterministic matrices in \({\mathbb {M}}_N({\mathbb {C}})\otimes {\mathbb {M}}_M({\mathbb {C}})\). Let P be a self-adjoint non-commutative polynomial. In Voiculescu (Int Math Res Notices 1:41–63, 1998), Voiculescu showed that the empirical measure of the eigenvalues of this polynomial evaluated in Haar unitary matrices and deterministic matrices converges towards a deterministic measure defined thanks to free probability theory. Now, let f be a smooth function. The main technical result of this paper is a precise bound of the difference between the expectation of $$\begin{aligned} \frac{1}{MN} {{\,\mathrm{Tr}\,}}_{{\mathbb {M}}_N({\mathbb {C}})}\otimes {{\,\mathrm{Tr}\,}}_{{\mathbb {M}}_M({\mathbb {C}})}\left( f(P(U^N\otimes I_M,Z^{NM})) \right) , \end{aligned}$$and its limit when N goes to infinity. If f is seven times differentiable, we show that it is bounded by \(M^2 \left\| f\right\| _{{\mathcal {C}}^6} \ln ^2(N)\times N^{-2}\). As a corollary we obtain a new proof with quantitative bounds of a result of Collins and Male which gives sufficient conditions for the operator norm of a polynomial evaluated in Haar unitary matrices and deterministic matrices to converge almost surely towards its free limit. Our result also holds in much greater generality. For instance, it allows to prove that if \(U^N\) and \(Y^{M_N}\) are independent and \(M_N=o(N^{1/3}\ln ^{-2/3}(N))\), then the norm of any polynomial in \((U^N\otimes I_{M_N}, I_N\otimes Y^{M_N})\) converges almost surely towards its free limit. Previous results required that \(M=M_N\) is constant.