# Multi-point functional central limit theorem for Wigner Matrices

Reker J. Multi-point functional central limit theorem for Wigner Matrices. arXiv, 10.48550/arXiv.2307.11028.

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https://doi.org/10.48550/arXiv.2307.11028
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Abstract

Consider the random variable $\mathrm{Tr}( f_1(W)A_1\dots f_k(W)A_k)$ where $W$ is an $N\times N$ Hermitian Wigner matrix, $k\in\mathbb{N}$, and choose (possibly $N$-dependent) regular functions $f_1,\dots, f_k$ as well as bounded deterministic matrices $A_1,\dots,A_k$. We give a functional central limit theorem showing that the fluctuations around the expectation are Gaussian. Moreover, we determine the limiting covariance structure and give explicit error bounds in terms of the scaling of $f_1,\dots,f_k$ and the number of traceless matrices among $A_1,\dots,A_k$, thus extending the results of [Cipolloni, Erdős, Schröder 2023] to products of arbitrary length $k\geq2$. As an application, we consider the fluctuation of $\mathrm{Tr}(\mathrm{e}^{\mathrm{i} tW}A_1\mathrm{e}^{-\mathrm{i} tW}A_2)$ around its thermal value $\mathrm{Tr}(A_1)\mathrm{Tr}(A_2)$ when $t$ is large and give an explicit formula for the variance.

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Date Published

2023-07-21

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arXiv

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### Cite this

Reker J. Multi-point functional central limit theorem for Wigner Matrices.

*arXiv*. doi:10.48550/arXiv.2307.11028Reker, J. (n.d.). Multi-point functional central limit theorem for Wigner Matrices.

*arXiv*. https://doi.org/10.48550/arXiv.2307.11028Reker, Jana. “Multi-Point Functional Central Limit Theorem for Wigner Matrices.”

*ArXiv*, n.d. https://doi.org/10.48550/arXiv.2307.11028.J. Reker, “Multi-point functional central limit theorem for Wigner Matrices,”

*arXiv*. .Reker, Jana. “Multi-Point Functional Central Limit Theorem for Wigner Matrices.”

*ArXiv*, doi:10.48550/arXiv.2307.11028.**All files available under the following license(s):**

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arXiv 2307.11028