Speaker: Phil Sosoe (Cornell University)

Title: Almost-optimal regularity conditions in the CLT for Wigner matrices

Abstract: We consider linear spectral statistics for test functions of low regularity and Wigner matrices with smooth entry distribution. We show that for functions in the Sobolev space $H^{1/2 + \epsilon}$ or the space $C^{1/2 + \epsilon}$ that are supported within the spectral bulk of the semicircle distribution, the variance remains bounded asymptotically. As a consequence, these linear spectral statistics have asymptotic Gaussian fluctuations with the same variance as in the CLT for functions of higher regularity, for any $\epsilon > 0$. This result is nearly optimal in the sense that the variance does remain bounded for functions in $H^{1/2}$, and was previously known only for matrices in Gaussian Unitary Ensemble.