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arXiv · 2204.03419

Almost-optimal bulk regularity conditions in the CLT for Wigner matrices

Benjamin Landon·Philippe Sosoe
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Abstract

We consider linear spectral statistics of the form $\mathrm{tr} ( \varphi (H))$ for test functions $\varphi$ of low regularity and Wigner matrices $H$ with smooth entry distribution. We show that for functions $\varphi$ in the Sobolev space $H^{1/2+\varepsilon}$ or the space $C^{1/2+\varepsilon}$, that are supported within the spectral bulk of the semicircle distribution, these linear spectral statistics have asymptotic Gaussian fluctuations with the same variance as in the CLT for functions of higher regularity, for any $\varepsilon >0$.

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Benjamin Landon, Philippe Sosoe. 2022-04-07. Almost-optimal bulk regularity conditions in the CLT for Wigner matrices. https://arxiv.org/abs/2204.03419

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