Related papers: Localization of One-Dimensional Random Band Matric…

We consider some random band matrices with band-width $N^\mu$ whose entries are independent random variables with distribution tail in $x^{-\alpha}$. We consider the largest eigenvalues and the associated eigenvectors and prove the…

We study large $N\times N$ power-law random band matrices $H=(H_{ij})$ with centered complex Gaussian entries, where the variances satisfy a power-law decay $\mathbb{E}|H_{ij}|^2\propto (|i-j|/W+1)^{-1-\alpha}$, for some exponent…

Consider an $ N \times N$ Hermitian one-dimensional random band matrix with band width $W > N^{1 / 2 + \frak c} $ for any $ {\frak c} > 0$. In the bulk of the spectrum and in the large $ N $ limit, we obtain the following results: (i) The…

It is shown that certain ensembles of random matrices with entries that vanish outside a band around the diagonal satisfy a localization condition on the resolvent which guarantees that eigenvectors have strong overlap with a vanishing…

We consider a class of random permutations of the interval $[-n,n]$, in which points are typically displaced a distance $O(W)$. We show the cycles are localized on the scale $W^3$, with an exponentially decaying tail bound. Analogous to…

Consider $N\times N$ symmetric one-dimensional random band matrices with general distribution of the entries and band width $W \geq N^{3/4+\varepsilon}$ for any $\varepsilon>0$. In the bulk of the spectrum and in the large $N$ limit, we…

We consider $N\times N$ Hermitian random band matrices $H=(H_{xy})$, whose entries are centered complex Gaussian random variables. The indices $x,y$ range over the $d$-dimensional discrete torus $(\mathbb Z/L\mathbb Z)^d$ with $d\in…

We study a random band matrix $H=(H_{xy})_{x,y}$ of dimension $N\times N$ with mean-zero complex Gaussian entries, where $x,y$ belong to the discrete torus $(\mathbb{Z}/\sqrt{N}\mathbb{Z})^{2}$. The variance profile…

We study random, symmetric $N \times N$ band matrices with a band of size $W$ and Bernoulli random variables as entries. This interpolates between nearest neighbour interaction $W = 1$ and Wigner matrices $W = N$. Eigenvectors are known to…

We consider a general class of symmetric or Hermitian random band matrices $H=(h_{xy})_{x,y \in \llbracket 1,N\rrbracket^d}$ in any dimension $d\ge 1$, where the entries are independent, centered random variables with variances…

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…

We present a novel block reduction method for the study of a general class of random band matrices (RBM) defined on the $d$-dimensional lattice $\mathbb{Z}_{L}^d:=\{1,2,\ldots,L\}^{d}$ for $d\in \{1,2\}$, with band width $W$ and an almost…

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, uniformly distributed random variables if $\abs{x-y}$ is less…

We consider Hermitian and symmetric random band matrices $H = (h_{xy})$ in $d \geq 1$ dimensions. The matrix entries $h_{xy}$, indexed by $x,y \in (\bZ/L\bZ)^d$, are independent, centred random variables with variances $s_{xy} = \E…

We consider Hermitian random band matrices $H=(h_{xy})$ on the $d$-dimensional lattice $(\mathbb Z/L\mathbb Z)^d$. The entries $h_{xy}$ are independent (up to Hermitian conditions) centered complex Gaussian random variables with variances…

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 show the eigenvectors of a Gaussian random band matrix are localized when the band width is less than the 1/4 power of the matrix size. Our argument is essentially an optimized version of Schenker's proof of the 1/8 exponent.

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…

We consider $N\times N$ self-adjoint Gaussian random matrices defined by an arbitrary deterministic sparsity pattern with $d$ nonzero entries per row. We show that such random matrices exhibit a canonical localization-delocalization…

Let $x \in S^{n-1}$ be a unit eigenvector of an $n \times n$ random matrix. This vector is delocalized if it is distributed roughly uniformly over the real or complex sphere. This intuitive notion can be quantified in various ways. In these…