Related papers: Bulk universality and quantum unique ergodicity fo…

We show that the fluctuations of the largest eigenvalue of a real symmetric or complex Hermitian Wigner matrix of size $N$ converge to the Tracy--Widom laws at a rate $O(N^{-1/3+\omega})$, as $N$ tends to infinity. For Wigner matrices this…

Consider $N\times N$ hermitian or symmetric random matrices $H$ with independent entries, where the distribution of the $(i,j)$ matrix element is given by the probability measure $\nu_{ij}$ with zero expectation and with variance…

Universality of eigenvalue spacings is one of the basic characteristics of random matrices. We give the precise meaning of universality and discuss the standard universality classes (sine, Airy, Bessel) and their appearance in unitary,…

This is a continuation of our earlier paper on the universality of the eigenvalues of Wigner random matrices. The main new results of this paper are an extension of the results in that paper from the bulk of the spectrum up to the edge. In…

We consider the least singular value of $M = R^* X T + U^* YV$, where $R,T,U,V$ are independent Haar-distributed unitary matrices and $X, Y$ are deterministic diagonal matrices. Under weak conditions on $X$ and $Y$, we show that the…

We consider random matrices of the form $H_N=A_N+U_N B_N U^*_N$, where $A_N$, $B_N$ are two $N$ by $N$ deterministic Hermitian matrices and $U_N$ is a Haar distributed random unitary matrix. We establish a universal Central Limit Theorem…

We study Hermitian random matrix models with an external source matrix which has equispaced eigenvalues, and with an external field such that the limiting mean density of eigenvalues is supported on a single interval as the dimension tends…

We prove universality for the fluctuations of the halting time for the Toda algorithm to compute the largest eigenvalue of real symmetric and complex Hermitian matrices. The proof relies on recent results on the statistics of the…

We study the eigenvalues of the covariance matrix $\frac{1}{n}M^*M$ of a large rectangular matrix $M=M_{n,p}=(\zeta_{ij})_{1\leq i\leq p;1\leq j\leq n}$ whose entries are i.i.d. random variables of mean zero, variance one, and having finite…

We consider the random matrix ensemble with an external source \[ \frac{1}{Z_n} e^{-n \Tr({1/2}M^2 -AM)} dM \] defined on $n\times n$ Hermitian matrices, where $A$ is a diagonal matrix with only two eigenvalues $\pm a$ of equal…

In this paper we consider $N \times N$ real generalized Wigner matrices whose entries are only assumed to have finite $(2 + \varepsilon)$-th moment for some fixed, but arbitrarily small, $\varepsilon > 0$. We show that the Stieltjes…

We study a class of Gaussian random band matrices of dimension $N \times N$ and band-width $W$. We show that delocalization holds for bulk eigenvectors and that quantum diffusion holds for the resolvent, all under the assumption that $W \gg…

We study the distribution of eigenvalues of almost-Hermitian random matrices associated with the classical Gaussian and Laguerre unitary ensembles. In the almost-Hermitian setting, which was pioneered by Fyodorov, Khoruzhenko and Sommers in…

We address the question of how the celebrated universality of local correlations for the real eigenvalues of Hermitian random matrices of size NxN can be extended to complex eigenvalues in the case of random matrices without symmetry.…

We study invariant random matrix ensembles \begin{equation*} \mathbb{P}_n(d M)=Z_n^{-1}\exp(-n\,tr(V(M)))\,d M \end{equation*} defined on complex Hermitian matrices $M$ of size $n\times n$, where $V$ is real analytic such that the…

We study the universality of the eigenvalue statistics of the covariance matrices $\frac{1}{n}M^* M$ where $M$ is a large $p\times n$ matrix obeying condition $\bf{C1}$. In particular, as an application, we prove a variant of universality…

Complex eigenvalues of random matrices $J=\text{GUE }+ i\gamma \diag (1, 0, \ldots, 0)$ provide the simplest model for studying resonances in wave scattering from a quantum chaotic system via a single open channel. It is known that in the…

In this series, we investigate quantum ergodicity at small scales for linear hyperbolic maps of the torus ("cat maps"). In Part I of the series, we prove quantum ergodicity at various scales. Let $N=1/h$, in which $h$ is the Planck…

We consider general self-adjoint polynomials in several independent random matrices whose entries are centered and have the same variance. We show that under certain conditions the local law holds up to the optimal scale, i.e., the…

We describe the resolvent approach for the rigorous study of the mescoscopic regime of Hermitian matrix spectra. We present results reflecting the universal behavior of the smoothed density of eigenvalue distribution of large random…