1,030 research outputs found
Delocalization and Diffusion Profile for Random Band Matrices
We consider Hermitian and symmetric random band matrices in dimensions. The matrix entries , 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 is negligible if exceeds the
band width . In one dimension we prove that the eigenvectors of are
delocalized if . We also show that the magnitude of the matrix
entries \abs{G_{xy}}^2 of the resolvent is self-averaging
and we compute \E \abs{G_{xy}}^2. We show that, as and , 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
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
Remarks on the derivation of Gross-Pitaevskii equation with magnetic Laplacian
The effective dynamics for a Bose-Einstein condensate in the regime of high
dilution and subject to an external magnetic field is governed by a magnetic
Gross-Pitaevskii equation. We elucidate the steps needed to adapt to the
magnetic case the proof of the derivation of the Gross-Pitaevskii equation
within the "projection counting" scheme
On the swap-distances of different realizations of a graphical degree sequence
One of the first graph theoretical problems which got serious attention
(already in the fifties of the last century) was to decide whether a given
integer sequence is equal to the degree sequence of a simple graph (or it is
{\em graphical} for short). One method to solve this problem is the greedy
algorithm of Havel and Hakimi, which is based on the {\em swap} operation.
Another, closely related question is to find a sequence of swap operations to
transform one graphical realization into another one of the same degree
sequence. This latter problem got particular emphases in connection of fast
mixing Markov chain approaches to sample uniformly all possible realizations of
a given degree sequence. (This becomes a matter of interest in connection of --
among others -- the study of large social networks.) Earlier there were only
crude upper bounds on the shortest possible length of such swap sequences
between two realizations. In this paper we develop formulae (Gallai-type
identities) for these {\em swap-distance}s of any two realizations of simple
undirected or directed degree sequences. These identities improves considerably
the known upper bounds on the swap-distances.Comment: to be publishe
The Linear Boltzmann Equation as the Low Density Limit of a Random Schrodinger Equation
We study the evolution of a quantum particle interacting with a random
potential in the low density limit (Boltzmann-Grad). The phase space density of
the quantum evolution defined through the Husimi function converges weakly to a
linear Boltzmann equation with collision kernel given by the full quantum
scattering cross section.Comment: 74 pages, 4 figures, (Final version -- typos corrected
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 vertices where every edge is chosen independently
and with probability . We rescale the matrix so that its bulk
eigenvalues are of order one. Under the assumption , 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 moments
Rate of Convergence Towards Semi-Relativistic Hartree Dynamics
We consider the semi-relativistic system of gravitating Bosons with
gravitation constant . The time evolution of the system is described by the
relativistic dispersion law, and we assume the mean-field scaling of the
interaction where and while fixed. In
the super-critical regime of large , we introduce the regularized
interaction where the cutoff vanishes as . We show that the
difference between the many-body semi-relativistic Schr\"{o}dinger dynamics and
the corresponding semi-relativistic Hartree dynamics is at most of order
for all , i.e., the result covers the sub-critical regime and
the super-critical regime. The dependence of the bound is optimal.Comment: 29 page
Relativistic Scott correction in self-generated magnetic fields
We consider a large neutral molecule with total nuclear charge 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
, where denotes the fine structure constant. We are
interested in the ground state energy in the simultaneous limit , such that is fixed. The
leading term in the energy asymptotics is independent of , it is given
by the Thomas-Fermi energy of order 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 . 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 , 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
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
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