1,294 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
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
Second-order corrections to mean-field evolution of weakly interacting Bosons, II
We study the evolution of a N-body weakly interacting system of Bosons. Our
work forms an extension of our previous paper I, in which we derived a
second-order correction to a mean-field evolution law for coherent states in
the presence of small interaction potential. Here, we remove the assumption of
smallness of the interaction potential and prove global existence of solutions
to the equation for the second-order correction. This implies an improved
Fock-space estimate for our approximation of the N-body state
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
Soft random solids and their heterogeneous elasticity
Spatial heterogeneity in the elastic properties of soft random solids is
examined via vulcanization theory. The spatial heterogeneity in the
\emph{structure} of soft random solids is a result of the fluctuations
locked-in at their synthesis, which also brings heterogeneity in their
\emph{elastic properties}. Vulcanization theory studies semi-microscopic models
of random-solid-forming systems, and applies replica field theory to deal with
their quenched disorder and thermal fluctuations. The elastic deformations of
soft random solids are argued to be described by the Goldstone sector of
fluctuations contained in vulcanization theory, associated with a subtle form
of spontaneous symmetry breaking that is associated with the
liquid-to-random-solid transition. The resulting free energy of this Goldstone
sector can be reinterpreted as arising from a phenomenological description of
an elastic medium with quenched disorder. Through this comparison, we arrive at
the statistics of the quenched disorder of the elasticity of soft random
solids, in terms of residual stress and Lam\'e-coefficient fields. In
particular, there are large residual stresses in the equilibrium reference
state, and the disorder correlators involving the residual stress are found to
be long-ranged and governed by a universal parameter that also gives the mean
shear modulus.Comment: 40 pages, 7 figure
Random graphs containing arbitrary distributions of subgraphs
Traditional random graph models of networks generate networks that are
locally tree-like, meaning that all local neighborhoods take the form of trees.
In this respect such models are highly unrealistic, most real networks having
strongly non-tree-like neighborhoods that contain short loops, cliques, or
other biconnected subgraphs. In this paper we propose and analyze a new class
of random graph models that incorporates general subgraphs, allowing for
non-tree-like neighborhoods while still remaining solvable for many fundamental
network properties. Among other things we give solutions for the size of the
giant component, the position of the phase transition at which the giant
component appears, and percolation properties for both site and bond
percolation on networks generated by the model.Comment: 12 pages, 6 figures, 1 tabl
Cavity Approach to the Random Solid State
The cavity approach is used to address the physical properties of random
solids in equilibrium. Particular attention is paid to the fraction of
localized particles and the distribution of localization lengths characterizing
their thermal motion. This approach is of relevance to a wide class of random
solids, including rubbery media (formed via the vulcanization of polymer
fluids) and chemical gels (formed by the random covalent bonding of fluids of
atoms or small molecules). The cavity approach confirms results that have been
obtained previously via replica mean-field theory, doing so in a way that sheds
new light on their physical origin.Comment: 4 pages, 2 figure
Growth of graph states in quantum networks
We propose a scheme to distribute graph states over quantum networks in the
presence of noise in the channels and in the operations. The protocol can be
implemented efficiently for large graph sates of arbitrary (complex) topology.
We benchmark our scheme with two protocols where each connected component is
prepared in a node belonging to the component and subsequently distributed via
quantum repeaters to the remaining connected nodes. We show that the fidelity
of the generated graphs can be written as the partition function of a classical
Ising-type Hamiltonian. We give exact expressions of the fidelity of the linear
cluster and results for its decay rate in random graphs with arbitrary
(uncorrelated) degree distributions.Comment: 16 pages, 7 figure
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
Quantum Diffusion and Eigenfunction Delocalization in a Random Band Matrix Model
We consider Hermitian and symmetric random band matrices in
dimensions. The matrix elements , indexed by , are independent, uniformly distributed random variables if \abs{x-y}
is less than the band width , and zero otherwise. We prove that the time
evolution of a quantum particle subject to the Hamiltonian is diffusive on
time scales . We also show that the localization length of an
arbitrarily large majority of the eigenvectors is larger than a factor
times the band width. All results are uniform in the size
\abs{\Lambda} of the matrix.Comment: Minor corrections, Sections 4 and 11 update
- …