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 [48] for the CUE and the sine process and the asymptotics formulae derived by Deift et al. [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 introduced in [19]. For the GUE kernel, we also discuss how to obtain the necessary sine-kernel asymptotics at mesoscopic scale by elementary means.

Subcritical Multiplicative Chaos for Regularized Counting Statistics from Random Matrix Theory

“…It turns out that the appearance of the H 1/2 -Gaussian noise is remarkably universal in the mesoscopic limit of one dimensional ensembles with random matrix type repulsion and this problem has attracted renewed interest in the last couple of years. For instance, the analogue of Soshnikov's CLT (1.9) was obtained for the GUE [28], more general invariant ensembles [12,46], Wigner matrices [23,47,34], β-ensembles [10,2] and for zeros of the Riemann zeta function [11,63]. It is likely the counterpart of Theorem 1.3 continues to hold for these models as well.…”

Incomplete Determinantal Processes: From Random Matrix to Poisson Statistics

“…Since it is the case for most ensembles, we shall also assume that the kernel K(z, w) is continuous on X × X. The cumulants method to analyze the asymptotic distribution of linear statistic of determinantal processes goes back to the work of Costin and Lebowitz, [17], for count statistics of the sine process and the general theory was developed by Soshnikov, [42,43,44], and subsequently applied to many different ensembles coming from random matrix theory, see for instance [40,39,2,14,15,31,34,35]. In this section, we show how to implement it to describe the asymptotics law of linear statistics of the incomplete ensembleΞ with correlation kernel pK(z, w) when 0 < p < 1.…”

“…The combinatorial method used in the previous section is well-suited to investigate the global properties of the processes Ξ andΞ but it is difficult to implement in the mesoscopic regime (in this case, polynomial linear statistics do not converge without normalization). Although there is a mesoscopic theory based on the asymptotics of the recurrence matrix (4.8) developed in [14], it is simpler to prove the mesoscopic counterpart of theorem 4.1 using the sine-kernel asymptotics of the correlation kernel K N V and the method of [34]. This reduces the proof to carefully apply the classical asymptotics of [20].…”

“…In the mesoscopic regime, the analysis is similar to the two-dimensional case and relies on the asymptotics of the correlation kernel K N V and the techniques introduced in [34]. Finally, C > 0 denotes a numerical constant which changes from line to line and, for any n ∈ N, we let for all x ∈ X n , dµ n (x) = dµ(x 1 ) · · · dµ(x n ).…”

A Functional Limit Theorem for the Sine-Process

“…For further developments see also works [21,25,26,7,8], where certain Central Limit Theorems were established for linear statistics with bounded variance, related to the marginal convergence D(ξ N g i ) → D(ξ g i ). More precisely, in [21,25] and [7] the Central Limit Theorems were proven for linear statistics of various orthogonal polynomial ensembles on mesoscopic scales. In [26] and [8] those were obtained for linear statistics of certain biorthogonal ensembles.…”