Mesoscopic fluctuations for unitary invariant ensembles

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 [48] for the CUE and the sine process and the asymptotics formulae derived by Deift et al. in [12]. For varying weights $e^{-N \tr V (H)}$, in the one-cut regime, we also provide estimates for the variance of linear statistics $\tr f(H)$ which are valid for a rather general function $f$. In particular, this implies that the characteristic polynomials of such Hermitian random matrices converge in a suitable regime to a regularized fractional Brownian motion with logarithmic correlations defined in [19]. For the GUE kernel, we also discuss how to obtain the necessary sine-kernel asymptotics at mesoscopic scale by elementary means.