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

Abstract

Let $X^N=(X_1^N,\dots ,X_d^N)$ be a d-tuple of $N\times N$ independent GUE random 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. 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 of $$\frac{1}{MN}\text{Tr}\left(f(P(X^N\otimes I_M,Z^{NM}))\right)$$ and its limit when $N$ goes to infinity. If $f$ is six times differentiable, we show that it is bounded by $M^2{\Vert f\Vert }_{{\mathcal{C}}^6}{N}^{-2}$. As a corollary we obtain a new proof of a result of Haagerup and Thorbjørnsen, later developed by Male, which gives sufficient conditions for the operator norm of a polynomial evaluated in $(X^N,Z^{NM},{Z^{NM}}^{\ast })$ to converge almost surely towards its free limit. Restricting ourselves to polynomials in independent GUE matrices, we give concentration estimates on the largest eingenvalue of these polynomials around their free limit. A direct consequence of these inequalities is that there exists some $\beta >0$ such that for any ${\epsilon }_1<3+\beta {)}^{-1}$ and ${\epsilon }_2<1/4$, almost surely for $N$ large enough, $$-\frac{1}{N^{{\epsilon }_1}}\le \Vert P(X^N)\Vert -\Vert P(x)\Vert \le \frac{1}{N^{{\epsilon }_2}}.$$ Finally if $X^N$ and $Y^{M_N}$ are independent and $M_N=o(N^{1/3})$, then almost surely, the norm of any polynomial in $(X^N\otimes I_{M_N},I_N\otimes Y^{M_N})$ converges almost surely towards its free limit. This result is an improvement of a Theorem of Pisier, who was himself using estimates from Haagerup and Thorbjørnsen, where $M_N$ had size $o(N^{1/4})$.

References in corpus (1)

• Second Order Freeness and Fluctuations of Random Matrices: II. Unitary Random Matrices

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