Asymptotic expansion of smooth functions in polynomials in deterministic matrices and iid GUE matrices

DOI10.1007/S00220-022-04551-2arXiv2011.04146MaRDI QIDQ6353272

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Publication date: 8 November 2020

Abstract: Let $X^N$ be a family of $NimesN$ independent GUE random matrices, $Z^N$ a family of deterministic matrices, $P$ a self-adjoint non-commutative polynomial, that is for any $N$, $P(X^N)$ is self-adjoint, $f$ a smooth function. We prove that for any $k$, if $f$ is smooth enough, there exist deterministic constants $alph{a}_i^P(f,Z^N)$ such that ${\displaystyle }$ mathbb{E}left[frac{1}{N} ext{Tr}left( f(P(X^N,Z^N)) ight) ight] = sum_{i=0}^k frac{alpha_i^P(f,Z^N)}{N^{2i}} + mathcal{O}(N^{-2k-2}) .${\displaystyle }$ Besides the constants $alph{a}_i^P(f,Z^N)$ 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, for all $i$, $alph{a}_i^P(f,Z^N)=0$. 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 from the spectrum of $P(x,Z^N)$.

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