Related papers: Mesoscopic linear statistics of Wigner matrices

We show that the fluctuations of the largest eigenvalue of any generalized Wigner matrix $H$ converge to the Tracy-Widom laws at a rate nearly $O(N^{-1/3})$, as the matrix dimension $N$ tends to infinity. We allow the variances of the…

We investigate the eigenvalues statistics of ensembles of normal random matrices when their order N tends to infinite. In the model the eigenvalues have uniform density within a region determined by a simple analytic polynomial curve. We…

In this article, we establish a near-optimal convergence rate for the CLT of linear eigenvalue statistics of Wigner matrices, in Kolmogorov-Smirnov distance. For all test functions $f\in C^5(\mathbb R)$, we show that the convergence rate is…

We prove the Wigner-Dyson-Mehta conjecture at fixed energy in the bulk of the spectrum for generalized symmetric and Hermitian Wigner matrices. Previous results concerning the universality of random matrices either require an averaging in…

We study the statistical regularity of Mather measures associated with $C^1$ perturbations of a Tonelli Lagrangian. When the unperturbed Mather measure is supported on a quasi-periodic torus with a Diophantine frequency, we establish…

By the continuous mapping theorem, if a sequence of $d$-dimensional random vectors $(\mathbf{W}_n)_{n\geq1}$ converges in distribution to a multivariate normal random variable $\Sigma^{1/2}\mathbf{Z}$, then the sequence of random variables…

We consider random Hermitian matrices with independent upper triangular entries. Wigner's semicircle law says that under certain additional assumptions, the empirical spectral distribution converges to the semicircle distribution. We…

We consider sequences of random variables of the type $S_n= n^{-1/2} \sum_{k=1}^n \{f(X_k)-\E[f(X_k)]\}$, $n\geq 1$, where $X=(X_k)_{k\in \Z}$ is a $d$-dimensional Gaussian process and $f: \R^d \rightarrow \R$ is a measurable function. It…

An invariant ensemble of $N\times N$ random matrices can be characterised by a joint distribution for eigenvalues $P(\lambda_1,\cdots,\lambda_N)$. The study of the distribution of linear statistics, i.e. of quantities of the form…

We consider the weak detection problem in a rank-one spiked Wigner data matrix where the signal-to-noise ratio is small so that reliable detection is impossible. We propose a hypothesis test on the presence of the signal by utilizing the…

We study linear spectral statistics of high dimensional sample covariance matrices in a regime where the empirical spectral distribution remains governed by the classical sample covariance law but the fluctuation theory is nonclassical. Our…

It is known that the fluctuations of suitable linear statistics of Haar distributed elements of the compact classical groups satisfy a central limit theorem. We show that if the corresponding test functions are sufficiently smooth, a rate…

This paper studies the winding of a continuously differentiable Gaussian stationary process $f:\mathbb{R}\to\mathbb{C}$ in the interval $[0,T]$. We give formulae for the mean and the variance of this random variable. The variance is shown…

We study the effective estimation of the diffusivity and Hurst parameter for the homogenized limit of a class of slow/fast systems. Depending on the system parameters, this limit solves a stochastic differential equation driven by either a…

We consider the singular vectors of any $m \times n$ submatrix of a rectangular $M \times N$ Gaussian matrix and study their asymptotic overlaps with those of the full matrix, in the macroscopic regime where $N \,/\, M\,$, $m \,/\, M$ as…

We compute spectra of large stochastic matrices $W$, defined on sparse random graphs, where edges $(i,j)$ of the graph are given positive random weights $W_{ij}>0$ in such a fashion that column sums are normalized to one. We compute spectra…

This paper studies the central limit theorems (CLTs) for linear spectral statistics (LSSs) of general sample covariance matrices, when the test functions belong to $C^3$, the class of functions with continuous third order derivatives. We…

We have studied statistical properties of the values of the Wigner function W(x) of 1D quantum maps on compact 2D phase space of finite area V. For this purpose we have defined a Wigner function probability distribution P(w) = (1/V) int…

Whittle-Mat\'ern fields are a recently introduced class of Gaussian processes on metric graphs, which are specified as solutions to a fractional-order stochastic differential equation. Unlike earlier covariance-based approaches for…

We study the minimal sample size N=N(n) that suffices to estimate the covariance matrix of an n-dimensional distribution by the sample covariance matrix in the operator norm, with an arbitrary fixed accuracy. We establish the optimal bound…