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

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Abstract

Let U_N = (U_N^1 ,. .. , U_N^p) be a d-tuple of N × N independent Haar unitary 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. In 1991, 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. 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 1/(MN) Tr( f (P (U_N ⊗ I_M , Z_{NM})) ) , and its limit when N goes to infinity. If f is seven times differentiable, we show that it is bounded by M^2 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 are independent and M = o(N 1/3), then the norm of any polynomial in (U_N ⊗ I_M N , I_N ⊗ Y_M) converges almost surely towards its free limit. Previous results required that M is bounded.
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Dates and versions

ensl-03053035 , version 1 (10-12-2020)

Identifiers

  • HAL Id : ensl-03053035 , version 1

Cite

Félix Parraud. On the operator norm of non-commutative polynomials in deterministic matrices and iid Haar unitary matrices. 2020. ⟨ensl-03053035⟩
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