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- W3103409668 abstract "Let $$X^N$$ be a family of $$Ntimes N$$ 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 $$iin mathbb {N}$$ . 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 $$P(x,Z^N)$$ ." @default.
- W3103409668 created "2020-11-23" @default.
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- W3103409668 date "2022-11-10" @default.
- W3103409668 modified "2023-09-24" @default.
- W3103409668 title "Asymptotic Expansion of Smooth Functions in Polynomials in Deterministic Matrices and iid GUE Matrices" @default.
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- W3103409668 doi "https://doi.org/10.1007/s00220-022-04551-2" @default.
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