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• W2995836609 abstract "Let $X^N = (X_1^N,dots, X^N_d)$ be a d-tuple of $Ntimes N$ 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 $$frac{1}{MN}text{Tr}left( f(P(X^Notimes I_M,Z^{NM})) right)$$ and its limit when $N$ goes to infinity. If $f$ is six times differentiable, we show that it is bounded by $M^2leftVert frightVert_{mathcal{C}^6}N^{-2}$. As a corollary we obtain a new proof of a result of Haagerup and Thorbjo rnsen, 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. 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 $varepsilon_1<3+beta)^{-1}$ and $varepsilon_2<1/4$, almost surely for $N$ large enough, $$-frac{1}{N^{varepsilon_1}} | P(X^N)| - leftVert P(x)rightVert leq frac{1}{N^{varepsilon_2}}.$$ 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 $(X^Notimes I_{M_N},I_Notimes Y^{M_N})$ 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 Thorbjo rnsen, where $M_N$ had size $o(N^{1/4})$." @default.
• W2995836609 created "2019-12-26" @default.
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• W2995836609 date "2022-01-01" @default.
• W2995836609 modified "2023-09-25" @default.
• W2995836609 title "On the operator norm of non-commutative polynomials in deterministic matrices and iid GUE matrices" @default.
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• W2995836609 doi "https://doi.org/10.4310/cjm.2022.v10.n1.a3" @default.
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