Delocalization and Diffusion Profile for Random Band Matrices

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 |h_{xy}|^2. We assume that $s_{xy}$ is negligible if $|x-y|$ exceeds the band width $W$. In one dimension we prove that the eigenvectors of $H$ are delocalized if $W\gg L^{4/5}$. We also show that the magnitude of the matrix entries \abs{G_{xy}}^2 of the resolvent $G=G(z)=(H-z)^{-1}$ is self-averaging and we compute \E \abs{G_{xy}}^2. We show that, as $L\to\infty$ and $W\gg L^{4/5}$, the behaviour of \E |G_{xy}|^2 is governed by a diffusion operator whose diffusion constant we compute. Similar results are obtained in higher dimensions

Relativistic Scott correction in self-generated magnetic fields

We consider a large neutral molecule with total nuclear charge $Z$ in a model with self-generated classical magnetic field and where the kinetic energy of the electrons is treated relativistically. To ensure stability, we assume that $Z \alpha < 2/\pi$, where $\alpha$ denotes the fine structure constant. We are interested in the ground state energy in the simultaneous limit $Z \rightarrow \infty$, $\alpha \rightarrow 0$ such that $\kappa=Z \alpha$ is fixed. The leading term in the energy asymptotics is independent of $\kappa$, it is given by the Thomas-Fermi energy of order $Z^{7/3}$ and it is unchanged by including the self-generated magnetic field. We prove the first correction term to this energy, the so-called Scott correction of the form $S(\alpha Z) Z^2$. The current paper extends the result of \cite{SSS} on the Scott correction for relativistic molecules to include a self-generated magnetic field. Furthermore, we show that the corresponding Scott correction function $S$, first identified in \cite{SSS}, is unchanged by including a magnetic field. We also prove new Lieb-Thirring inequalities for the relativistic kinetic energy with magnetic fields.Comment: Small typos corrected, new references adde

Local Eigenvalue Density for General MANOVA Matrices

We consider random n\times n matrices of the form (XX*+YY*)^{-1/2}YY*(XX*+YY*)^{-1/2}, where X and Y have independent entries with zero mean and variance one. These matrices are the natural generalization of the Gaussian case, which are known as MANOVA matrices and which have joint eigenvalue density given by the third classical ensemble, the Jacobi ensemble. We show that, away from the spectral edge, the eigenvalue density converges to the limiting density of the Jacobi ensemble even on the shortest possible scales of order 1/n (up to \log n factors). This result is the analogue of the local Wigner semicircle law and the local Marchenko-Pastur law for general MANOVA matrices.Comment: Several small changes made to the tex

Strong polarized relations for the continuum

We prove that the strong polarized relation for the continuum holds for $\aleph_0$ and for every supercompact cardinal. We use iteration of Mathias forcing.Comment: 9 page

Diffusion of wave packets in a Markov random potential

We consider the evolution of a tight binding wave packet propagating in a time dependent potential. If the potential evolves according to a stationary Markov process, we show that the square amplitude of the wave packet converges, after diffusive rescaling, to a solution of a heat equation.Comment: 19 pages, acknowledgments added and typos correcte

Diffusive propagation of wave packets in a fluctuating periodic potential

We consider the evolution of a tight binding wave packet propagating in a fluctuating periodic potential. If the fluctuations stem from a stationary Markov process satisfying certain technical criteria, we show that the square amplitude of the wave packet after diffusive rescaling converges to a superposition of solutions of a heat equation.Comment: 13 pages (v2: added a paragraph on the history of the problem, added some references, correct a few typos; v3 minor corrections, added keywords and subject classes

Spectral Statistics of Erd{\H o}s-R\'enyi Graphs II: Eigenvalue Spacing and the Extreme Eigenvalues

We consider the ensemble of adjacency matrices of Erd{\H o}s-R\'enyi random graphs, i.e.\ 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 eigenvalues are of order one. Under the assumption $p N \gg N^{2/3}$, we prove the universality of eigenvalue distributions both in the bulk and at the edge of the spectrum. More precisely, we prove (1) that the eigenvalue spacing of the Erd{\H o}s-R\'enyi graph in the bulk of the spectrum has the same distribution as that of the Gaussian orthogonal ensemble; and (2) that the second largest eigenvalue of the Erd{\H o}s-R\'enyi graph has the same distribution as the largest eigenvalue of the Gaussian orthogonal ensemble. As an application of our method, we prove the bulk universality of generalized Wigner matrices under the assumption that the matrix entries have at least $4 + \epsilon$ moments

Relaxation dynamics of maximally clustered networks

We study the relaxation dynamics of fully clustered networks (maximal number of triangles) to an unclustered state under two different edge dynamics---the double-edge swap, corresponding to degree-preserving randomization of the configuration model, and single edge replacement, corresponding to full randomization of the Erd\H{o}s--R\'enyi random graph. We derive expressions for the time evolution of the degree distribution, edge multiplicity distribution and clustering coefficient. We show that under both dynamics networks undergo a continuous phase transition in which a giant connected component is formed. We calculate the position of the phase transition analytically using the Erd\H{o}s--R\'enyi phenomenology

Efficiency of informational transfer in regular and complex networks

We analyze the process of informational exchange through complex networks by measuring network efficiencies. Aiming to study non-clustered systems, we propose a modification of this measure on the local level. We apply this method to an extension of the class of small-worlds that includes {\it declustered} networks, and show that they are locally quite efficient, although their clustering coefficient is practically zero. Unweighted systems with small-world and scale-free topologies are shown to be both globally and locally efficient. Our method is also applied to characterize weighted networks. In particular we examine the properties of underground transportation systems of Madrid and Barcelona and reinterpret the results obtained for the Boston subway network.Comment: 10 pages and 9 figure

Spacings and pair correlations for finite Bernoulli convolutions

We consider finite Bernoulli convolutions with a parameter $1/2 < r < 1$ supported on a discrete point set, generically of size $2^N$. These sequences are uniformly distributed with respect to the infinite Bernoulli convolution measure $\nu_r$, as $N$ tends to infinity. Numerical evidence suggests that for a generic $r$, the distribution of spacings between appropriately rescaled points is Poissonian. We obtain some partial results in this direction; for instance, we show that, on average, the pair correlations do not exhibit attraction or repulsion in the limit. On the other hand, for certain algebraic $r$ the behavior is totally different.Comment: 17 pages, 6 figure