2020/05/28 by Félix Parraud, Parraud, Félix · 3 citations
Mathematics · #Advanced Topics in Algebra #FOS: Mathematics #Operator Algebras (math.OA) #Probability (math.PR) #Random Matrices and Applications #Spectral Theory in Mathematical Physics
paper · pdf · doi:10.48550/arxiv.2005.13834
openalex publication_date 2020/05/28 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Let UN = (U1N,…, UNp) be a d-tuple of N× N independent Haar unitary matrices and ZNM be any family of deterministic matrices in \mathbbMN(ℂ)⊗ \mathbbMM(ℂ). Let P be a self-adjoint non-commutative polynomial. In 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. 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(UN⊗ IM,ZNM)) ) , and its limit when N goes to infinity. If f is seven times differentiable, we show that it is bounded by M2 \Vert f\VertC7 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. Actually we show that if UN and YMN are independent and MN = o(N1/3), then almost surely, the norm of any polynomial in (UN⊗ IMN, IN⊗ YMN) converges almost surely towards its free limit.