Related papers: From non-ergodic eigenvectors to local resolvent s…
The article reviews the physics of Anderson localization on random regular graphs (RRG) and its connections to many-body localization (MBL) in disordered interacting systems. Properties of eigenstate and energy level correlations in…
We consider $N\times N$ symmetric or hermitian random matrices with independent, identically distributed entries where the probability distribution for each matrix element is given by a measure $\nu$ with a subexponential decay. We prove…
We perform an extensive investigation of the localization properties of the eigenmodes of the Laplace and adjacency matrix for one-dimensional random geometric graphs. We evaluate the density of states, the probability distribution of the…
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 present a non perturbative calculation technique providing the mixed moments of the overlaps between the eigenvectors of two large quantum Hamiltonians: $\hat{H}_0$ and $\hat{H}_0+\hat{W}$, where $\hat{H}_0$ is deterministic and…
Eigenvectors associated with non-degenerate eigenvalues are shown to correspond to columns of the adjugate of the characteristic matrix. Degenerate eigenvalues are associated with eigenvectors that correspond to reduced complement tensors…
The extreme-value statistics of the entanglement spectrum in disordered spin chains possessing a many-body localization transition is examined. It is expected that eigenstates in the metallic or ergodic phase, behave as random states and…
Our study connects the physics of disordered integer-dimensional systems and regular self-similar objects by studying spectral properties of fractal agglomerates with tunable dimension. The latter is controlled by parameter $\alpha$ of the…
Basing on our recent results on the $1/n$-expansion in unitary invariant random matrix ensembles, known as matrix models, we prove that the local eigenvalue statistic, arising in a certain neighborhood of the edges of the support of the…
We consider the eigenvectors of the principal minor of dimension $n< N$ of the Dyson Brownian motion in $\mathbb{R}^{N}$ and investigate their asymptotic overlaps with the eigenvectors of the full matrix in the limit of large dimension. We…
Domain specific localization of eigenstates has been a persistent observation for systems with local symmetries. The underlying mechanism for this localization behaviour has however remained elusive. We provide here an analysis of locally…
The diagonalization of matrices may be the top priority in the application of modern physics. In this paper, we numerically demonstrate that, for real symmetric random matrices with non-positive off-diagonal elements, a universal scaling…
This is the second part of a three part series abut delocalization for band matrices. In this paper, we consider a general class of $N\times N$ random band matrices $H=(H_{ij})$ whose entries are centered random variables, independent up to…
The non-backtracking operator of a graph is a powerful tool in spectral graph theory and random matrix theory. Most existing results for the non-backtracking operator of a random graph concern only eigenvalues or top eigenvectors. In this…
We consider the ensemble of adjacency matrices of Erd\H{o}s-R\'{e}nyi random graphs, that is, graphs on $N$ vertices where every edge is chosen independently and with probability $p\equiv p(N)$. We rescale the matrix so that its bulk…
We prove improved bounds on how localized an eigenvector of a high girth regular graph can be, and present examples showing that these bounds are close to sharp. This study was initiated by Brooks and Lindenstrauss (2009) who relied on the…
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…
Consider a random regular graph of fixed degree $d$ with $n$ vertices. We study spectral properties of the adjacency matrix and of random Schr\"odinger operators on such a graph as $n$ tends to infinity. We prove that the integrated density…
Gaussian Rosenzweig-Porter (GRP) random matrix ensemble is the only one in which the robust multifractal phase and ergodic transition have a status of a mathematical theorem. Yet, this phase in GRP model is oversimplified: the spectrum of…
The Eigenstate Thermalization Hypothesis (ETH) was developed as a framework for understanding how the principles of statistical mechanics emerge in the long-time limit of isolated quantum many-body systems. Since then, ETH has shifted the…