Related papers: Local deformed semicircle law and complete delocal…

We consider random $d$-regular graphs on $N$ vertices, with degree $d$ at least $(\log N)^4$. We prove that the Green's function of the adjacency matrix and the Stieltjes transform of its empirical spectral measure are well approximated by…

We consider the local eigenvalue distribution of large self-adjoint $N\times N$ random matrices $\mathbf{H}=\mathbf{H}^*$ with centered independent entries. In contrast to previous works the matrix of variances $s_{ij} = \mathbb{E}\,…

We prove that a very general class of $N\times N$ Hermitian random band matrices is in the delocalized phase when the band width $W$ exceeds the critical threshold, $W\gg \sqrt{N}$. In this regime, we show that, in the bulk spectrum, the…

It is a classical result of Wigner that for an hermitian matrix with independent entries on and above the diagonal, the mean empirical eigenvalue distribution converges weakly to the semicircle law as matrix size tends to infinity. In this…

We study the universality of spectral statistics of large random matrices. We consider $N\times N$ symmetric, hermitian or quaternion self-dual random matrices with independent, identically distributed entries (Wigner matrices) where the…

We prove a local law and eigenvector delocalization for general Wigner-type matrices. Our methods allow us to get the best possible interval length and optimal eigenvector delocalization in the dense case, and the first results of such kind…

We study the eigenvectors of generalized Wigner matrices with subexponential entries and prove that they delocalize at the optimal rate with overwhelming probability. We also prove high probability delocalization bounds with sharp…

We consider $N\times N$ Hermitian random matrices with independent identically distributed entries (Wigner matrices). The matrices are normalized so that the average spacing between consecutive eigenvalues is of order $1/N$. Under suitable…

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…

We study $N \times N$ random band matrices $H = (H_{xy})$ with mean-zero complex Gaussian entries, where $x,y$ lie on the discrete torus $(\mathbb{Z} / \sqrt[d]{N} \mathbb{Z})^d$ in dimensions $d \ge 3$. The variance profile satisfies…

This paper studies the delocalized regime of an ultrametric random operator whose independent entries have variances decaying in a suitable hierarchical metric on $\mathbb{N}$. When the decay-rate of the off-diagonal variances is…

In this text, we consider an N by N random matrix X such that all but o(N) rows of X have W non identically zero entries, the other rows having lass than $W$ entries (such as, for example, standard or cyclic band matrices). We always…

Consider $N\times N$ Hermitian or symmetric random matrices $H$ where the distribution of the $(i,j)$ matrix element is given by a probability measure $\nu_{ij}$ with a subexponential decay. Let $\sigma_{ij}^2$ be the variance for the…

We prove that a class of weakly perturbed Hamiltonians of the form $H_\lambda = H_0 + \lambda W$, with $W$ being a Wigner matrix, exhibits prethermalization. That is, the time evolution generated by $H_\lambda$ relaxes to its ultimate…

We prove that an n by n random matrix G with independent entries is completely delocalized. Suppose the entries of G have zero means, variances uniformly bounded below, and a uniform tail decay of exponential type. Then with high…

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…

We consider a general class of $N\times N$ random matrices whose entries $h_{ij}$ are independent up to a symmetry constraint, but not necessarily identically distributed. Our main result is a local semicircle law which improves previous…

We prove a general local law for Wigner matrices which optimally handles observables of arbitrary rank and thus it unifies the well-known averaged and isotropic local laws. As an application, we prove that the quadratic forms of a general…

We prove an optimal order delocalization estimate for the eigenvectors of general $N \times N$ non-Hermitian matrices $X$: $\| {\bf v } \|_\infty \leq C \sqrt{\frac{\log N}{N}}$ with very high probability, for any right or left eigenvector…

We consider Hermitian and symmetric random band matrices $H$ in $d \geq 1$ dimensions. The matrix elements $H_{xy}$, indexed by $x,y \in \Lambda \subset \Z^d$, are independent and their variances satisfy $\sigma_{xy}^2:=\E \abs{H_{xy}}^2 =…