Related papers: Poisson statistics of eigenvalues in the hierarchi…

We describe some numerical experiments which determine the degree of spectral instability of medium size randomly generated matrices which are far from self-adjoint. The conclusion is that the eigenvalues are likely to be intrinsically…

The one-dimensional (1d) Anderson model (AM) has statistical anomalies at any rational point $f=2a/\lambda_{E}$, where $a$ is the lattice constant and $\lambda_{E}$ is the de Broglie wavelength. We develop a regular approach to anomalous…

We consider random Schr\"{o}dinger operators on $\ell^2(\mathbb{Z}^d)$ when the distribution of single site potentials is $\alpha$-H\"{o}lder continuous ($0<\alpha\leq 1$). In localized regime we study the distribution of eigenfunctions…

In this paper, we study eigenvalue fluctuations of the finite volume Anderson model in the mesoscopic scale. We carry out this study in a regime of exponential localization and prove a central limit theorem for the eigenvalue counting…

Let $(X,d)$ be a locally compact separable ultrametric space. Given a measure $m$ on $X$ and a function $C$ defined on the set $\mathcal{B}$ of all balls $B\subset X$ we consider the hierarchical Laplacian $L=L_{C}$. The operator $L$ acts…

The distribution of the ratios of consecutive eigenvalue spacings of random matrices has emerged as an important tool to study spectral properties of many-body systems. This article numerically investigates the eigenvalue ratios…

We show that the spacing between eigenvalues of the discrete 1D Hamiltonian with arbitrary potentials which are bounded, and with Dirichlet or Neumann Boundary Conditions is bounded away from zero. We prove an explicit lower bound, given by…

We have calculated wave functions and matrix elements of the dipole operator in the two- and three-dimensional Anderson model of localization and have studied their statistical properties in the limit of weak disorder. In particular, we…

We consider a $p$-dimensional time series where the dimension $p$ increases with the sample size $n$. The resulting data matrix $X$ follows a stochastic volatility model: each entry consists of a positive random volatility term multiplied…

Products of random matrices associated to one-dimensional random media satisfy a central limit theorem assuring convergence to a gaussian centered at the Lyapunov exponent. The hypothesis of single parameter scaling states that its variance…

We analyze the eigenvalue statistics of the staggered Dirac operator above $T_{c}$ in QCD with 2+1 flavors of dynamical quarks. We use physical quark masses in our simulations. We compare the eigenvalue statistics from several parts of the…

We prove localization and probabilistic bounds on the minimum level spacing for a random block Anderson model without monotonicity. Using a sequence of narrowing energy windows and associated Schur complements, we obtain detailed…

The spectrum of exponents of the transfer matrix provides the localization lengths of Anderson's model for a particle in a lattice with disordered potential. I show that a duality identity for determinants and Jensen's identity for…

We study the level-statistics of a disordered system undergoing the Anderson type metal-insulator transition. The disordered Hamiltonian is a sparse random matrix in the site representation and the statistics is obtained by taking an…

We consider the adjacency matrix $A$ of the Erd\H{o}s-R\'enyi graph on $N$ vertices with edge probability $d/N$. For $(\log \log N)^4 \ll d \lesssim \log N$, we prove that the eigenvalues near the spectral edge form asymptotically a Poisson…

We develop a theory which describes the behaviour of eigenvalues of a class of one-dimensional random non-Hermitian operators introduced recently by Hatano and Nelson. Under general assumptions on random parameters we prove that the…

We consider disordered Hamiltonians given by the Laplace operator subject to arbitrary random self-adjoint singular perturbations supported on random discrete subsets of the real line. Under minimal assumptions on the type of disorder, we…

We demonstrate that by considering disordered single-particle Hamiltonians (or their random matrix versions) on ultrametric spaces one can generate an interesting class of models exhibiting Anderson metal-insulator transition. We use the…

Using the level--spacing distribution and the total probability function of the numbers of levels in a given energy interval we analyze the crossover of the level statistics between the delocalized and the localized regimes. By numerically…

Contrary to prevailing notion we find that the spectrum associated with the extended states in a complex system may belong to the Poisson universality class if the system is subjected to a specific set of constraints. Our results are based…