Mesoscopic fluctuations for unitary invariant ensembles

Abstract

Considering a determinantal point process on the real line, we establish a connection between the sine-kernel asymptotics for the correlation kernel and the CLT for mesoscopic linear statistics. This implies universality of mesoscopic fluctuations for a large class of unitary invariant Hermitian ensembles. In particular, this shows that the support of the equilibrium measure need not be connected in order to see Gaussian fluctuations at mesoscopic scales. Our proof is based on the cumulants computations introduced in [45] for the CUE and the sine process and the asymptotic formulae derived by Deift et al. [13]. For varying weights $e^{-N \operatorname{Tr} V (\mathrm{H} )}$, in the one-cut regime, we also provide estimates for the variance of linear statistics $\operatorname{Tr} f(\mathrm{H} )$ which are valid for a rather general function $f$. In particular, this implies that the logarithm of the absolute value of the characteristic polynomials of such Hermitian random matrices converges in a suitable regime to a regularized fractional Brownian motion with logarithmic correlations introduced in [17]. For the GUE and Jacobi ensembles, we also discuss how to obtain the necessary sine-kernel asymptotics at mesoscopic scale by elementary means.