52 research outputs found
Sharpness of the phase transition and exponential decay of the subcritical cluster size for percolation on quasi-transitive graphs
We study homogeneous, independent percolation on general quasi-transitive
graphs. We prove that in the disorder regime where all clusters are finite
almost surely, in fact the expectation of the cluster size is finite. This
extends a well-known theorem by Menshikov and Aizenman & Barsky to all
quasi-transitive graphs. Moreover we deduce that in this disorder regime the
cluster size distribution decays exponentially, extending a result of Aizenman
& Newman. Our results apply to both edge and site percolation, as well as long
range (edge) percolation. The proof is based on a modification of the Aizenman
& Barsky method.Comment: Latex 2e; 25 pages (a4wide); small editorial corrections; one
reference adde
Localization via fractional moments for models on with single-site potentials of finite support
One of the fundamental results in the theory of localization for discrete
Schr\"odinger operators with random potentials is the exponential decay of
Green's function and the absence of continuous spectrum. In this paper we
provide a new variant of these results for one-dimensional alloy-type
potentials with finitely supported sign-changing single-site potentials using
the fractional moment method.Comment: LaTeX-file, 26 pages with 2 LaTeX figure
Lifshitz tails for alloy type models in a constant magnetic field
In this note, we study Lifshitz tails for a 2D Landau Hamiltonian perturbed
by a random alloy-type potential constructed with single site potentials
decaying at least at a Gaussian speed. We prove that, if the Landau level stays
preserved as a band edge for the perturbed Hamiltonian, at the Landau levels,
the integrated density of states has a Lifshitz behavior of the type
Low lying spectrum of weak-disorder quantum waveguides
We study the low-lying spectrum of the Dirichlet Laplace operator on a
randomly wiggled strip. More precisely, our results are formulated in terms of
the eigenvalues of finite segment approximations of the infinite waveguide.
Under appropriate weak-disorder assumptions we obtain deterministic and
probabilistic bounds on the position of the lowest eigenvalue. A Combes-Thomas
argument allows us to obtain so-called 'initial length scale decay estimates'
at they are used in the proof of spectral localization using the multiscale
analysis.Comment: Accepted for publication in Journal of Statistical Physics
http://www.springerlink.com/content/0022-471
Anderson localization for a class of models with a sign-indefinite single-site potential via fractional moment method
A technically convenient signature of Anderson localization is exponential
decay of the fractional moments of the Green function within appropriate energy
ranges. We consider a random Hamiltonian on a lattice whose randomness is
generated by the sign-indefinite single-site potential, which is however
sign-definite at the boundary of its support. For this class of Anderson
operators we establish a finite-volume criterion which implies that above
mentioned the fractional moment decay property holds. This constructive
criterion is satisfied at typical perturbative regimes, e. g. at spectral
boundaries which satisfy 'Lifshitz tail estimates' on the density of states and
for sufficiently strong disorder. We also show how the fractional moment method
facilitates the proof of exponential (spectral) localization for such random
potentials.Comment: 29 pages, 1 figure, to appear in AH
Wegner estimate for discrete alloy-type models
We study discrete alloy-type random Schr\"odinger operators on
. Wegner estimates are bounds on the average number of
eigenvalues in an energy interval of finite box restrictions of these types of
operators. If the single site potential is compactly supported and the
distribution of the coupling constant is of bounded variation a Wegner estimate
holds. The bound is polynomial in the volume of the box and thus applicable as
an ingredient for a localisation proof via multiscale analysis.Comment: Accepted for publication in AHP. For an earlier version see
http://www.ma.utexas.edu/mp_arc-bin/mpa?yn=09-10
Novel Modifications of Parallel Jacobi Algorithms
We describe two main classes of one-sided trigonometric and hyperbolic
Jacobi-type algorithms for computing eigenvalues and eigenvectors of Hermitian
matrices. These types of algorithms exhibit significant advantages over many
other eigenvalue algorithms. If the matrices permit, both types of algorithms
compute the eigenvalues and eigenvectors with high relative accuracy.
We present novel parallelization techniques for both trigonometric and
hyperbolic classes of algorithms, as well as some new ideas on how pivoting in
each cycle of the algorithm can improve the speed of the parallel one-sided
algorithms. These parallelization approaches are applicable to both
distributed-memory and shared-memory machines.
The numerical testing performed indicates that the hyperbolic algorithms may
be superior to the trigonometric ones, although, in theory, the latter seem
more natural.Comment: Accepted for publication in Numerical Algorithm
Lp-approximation of the integrated density of states for Schrödinger operators with finite local complexity
A GPU-based hyperbolic SVD algorithm
A one-sided Jacobi hyperbolic singular value decomposition (HSVD) algorithm,
using a massively parallel graphics processing unit (GPU), is developed. The
algorithm also serves as the final stage of solving a symmetric indefinite
eigenvalue problem. Numerical testing demonstrates the gains in speed and
accuracy over sequential and MPI-parallelized variants of similar Jacobi-type
HSVD algorithms. Finally, possibilities of hybrid CPU--GPU parallelism are
discussed.Comment: Accepted for publication in BIT Numerical Mathematic
Local Wegner and Lifshitz tails estimates for the density of states for continuous random Schr\"odinger operators
We introduce and prove local Wegner estimates for continuous generalized
Anderson Hamiltonians, where the single-site random variables are independent
but not necessarily identically distributed. In particular, we get Wegner
estimates with a constant that goes to zero as we approach the bottom of the
spectrum. As an application, we show that the (differentiated) density of
states exhibits the same Lifshitz tails upper bound as the integrated density
of states.Comment: Revised with new titl
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