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Asymptotic Expansion of Smooth Functions in Polynomials in Deterministic Matrices and iid GUE Matrices

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Abstract

Let \(X^N\) be a family of \(N\times N\) independent GUE random matrices, \(Z^N\) a family of deterministic matrices, P a self-adjoint noncommutative polynomial, that is for any N, \(P(X^N,Z^N)\) is self-adjoint, f a smooth function. We prove that for any k, if f is smooth enough, there exist deterministic constants \(\alpha _i^P(f,Z^N)\) such that

$$\begin{aligned} \mathbb {E}\left[ \frac{1}{N}\text {Tr}\left( f(P(X^N,Z^N)) \right) \right] \ =\ \sum _{i=0}^k \frac{\alpha _i^P(f,Z^N)}{N^{2i}}\ +\ \mathcal {O}(N^{-2k-2}) . \end{aligned}$$

Besides, the constants \(\alpha _i^P(f,Z^N)\) are built explicitly with the help of free probability. In particular, if x is a free semicircular system, then when the support of f and the spectrum of \(P(x,Z^N)\) are disjoint, \(\alpha _i^P(f,Z^N)=0\) for all \(i\in \mathbb {N}\). As a corollary, we prove that given \(\alpha <1/2\), for N large enough, every eigenvalue of \(P(X^N,Z^N)\) is \(N^{-\alpha }\)-close to the spectrum of \(P(x,Z^N)\).

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Acknowledgements

The author would like to thanks his PhD supervisors Benoît Collins and Alice Guionnet for proofreading this paper and their continuous help, as well as Mikael de la Salle for helpful discussion. The author was partially supported by a MEXT JASSO fellowship and Labex Milyon (ANR-10-LABX-0070) of Université de Lyon.

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  1. Université de Lyon, ENSL, UMPA, 46 allée d’Italie, 69007, Lyon, France

    Félix Parraud

  2. Department of Mathematics, Graduate School of Science, Kyoto University, Kyoto, 606-8502, Japan

    Félix Parraud

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Parraud, F. Asymptotic Expansion of Smooth Functions in Polynomials in Deterministic Matrices and iid GUE Matrices. Commun. Math. Phys. 399, 249–294 (2023). https://doi.org/10.1007/s00220-022-04551-2

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