1,904 research outputs found
Uniform Poincare inequalities for unbounded conservative spin systems: The non-interacting case
We prove a uniform Poincare' inequality for non-interacting unbounded spin
systems with a conservation law, when the single-site potential is a bounded
perturbation of a convex function. The result is then applied to
Ginzburg-Landau processes to show diffusive scaling of the associated spectral
gap.Comment: 19 pages, revised version, to appear in Stoch. Proc. App
Large deviations of empirical neighborhood distribution in sparse random graphs
Consider the Erd\H{o}s-Renyi random graph on n vertices where each edge is
present independently with probability c/n, with c>0 fixed. For large n, a
typical random graph locally behaves like a Galton-Watson tree with Poisson
offspring distribution with mean c. Here, we study large deviations from this
typical behavior within the framework of the local weak convergence of finite
graph sequences. The associated rate function is expressed in terms of an
entropy functional on unimodular measures and takes finite values only at
measures supported on trees. We also establish large deviations for other
commonly studied random graph ensembles such as the uniform random graph with
given number of edges growing linearly with the number of vertices, or the
uniform random graph with given degree sequence. To prove our results, we
introduce a new configuration model which allows one to sample uniform random
graphs with a given neighborhood distribution, provided the latter is supported
on trees. We also introduce a new class of unimodular random trees, which
generalizes the usual Galton Watson tree with given degree distribution to the
case of neighborhoods of arbitrary finite depth. These generalized Galton
Watson trees turn out to be useful in the analysis of unimodular random trees
and may be considered to be of interest in their own right.Comment: 58 pages, 5 figure
Asymmetric diffusion and the energy gap above the 111 ground state of the quantum XXZ model
We consider the anisotropic three dimensional XXZ Heisenberg ferromagnet in a
cylinder with axis along the 111 direction and boundary conditions that induce
ground states describing an interface orthogonal to the cylinder axis. Let
be the linear size of the basis of the cylinder. Because of the breaking of the
continuous symmetry around the axis, the Goldstone theorem implies
that the spectral gap above such ground states must tend to zero as . In \cite{BCNS} it was proved that, by perturbing in a sub--cylinder
with basis of linear size the interface ground state, it is possible
to construct excited states whose energy gap shrinks as . Here we prove
that, uniformly in the height of the cylinder and in the location of the
interface, the energy gap above the interface ground state is bounded from
below by . We prove the result by first mapping the
problem into an asymmetric simple exclusion process on and then by
adapting to the latter the recursive analysis to estimate from below the
spectral gap of the associated Markov generator developed in \cite{CancMart}.
Along the way we improve some bounds on the equivalence of ensembles already
discussed in \cite{BCNS} and we establish an upper bound on the density of
states close to the bottom of the spectrum.Comment: 48 pages, latex2e fil
A large deviation principle for Wigner matrices without Gaussian tails
We consider Hermitian matrices with i.i.d. entries whose
tail probabilities behave like
for some and . We establish a large deviation principle
for the empirical spectral measure of with speed
with a good rate function that is finite only if is of the form
for some probability measure on
, where denotes the free convolution and
is Wigner's semicircle law. We obtain explicit expressions
for in terms of the th moment of
. The proof is based on the analysis of large deviations for the empirical
distribution of very sparse random rooted networks.Comment: Published in at http://dx.doi.org/10.1214/13-AOP866 the Annals of
Probability (http://www.imstat.org/aop/) by the Institute of Mathematical
Statistics (http://www.imstat.org
Phase ordering after a deep quench: the stochastic Ising and hard core gas models on a tree
Consider a low temperature stochastic Ising model in the phase coexistence
regime with Markov semigroup . A fundamental and still largely open
problem is the understanding of the long time behavior of \d_\h P_t when the
initial configuration \h is sampled from a highly disordered state
(e.g. a product Bernoulli measure or a high temperature Gibbs measure).
Exploiting recent progresses in the analysis of the mixing time of Monte Carlo
Markov chains for discrete spin models on a regular -ary tree \Tree^b, we
tackle the above problem for the Ising and hard core gas (independent sets)
models on \Tree^b. If is a biased product Bernoulli law then, under
various assumptions on the bias and on the thermodynamic parameters, we prove
-almost sure weak convergence of \d_\h P_t to an extremal Gibbs measure
(pure phase) and show that the limit is approached at least as fast as a
stretched exponential of the time . In the context of randomized algorithms
and if one considers the Glauber dynamics on a large, finite tree, our results
prove fast local relaxation to equilibrium on time scales much smaller than the
true mixing time, provided that the starting point of the chain is not taken as
the worst one but it is rather sampled from a suitable distribution.Comment: 35 page
Convergence to equilibrium for a directed (1+d)-dimensional polymer
We consider a flip dynamics for directed (1+d)-dimensional lattice paths with
length L. The model can be interpreted as a higher dimensional version of the
simple exclusion process, the latter corresponding to the case d=1. We prove
that the mixing time of the associated Markov chain scales like L^2\log L up to
a d-dependent multiplicative constant. The key step in the proof of the upper
bound is to show that the system satisfies a logarithmic Sobolev inequality on
the diffusive scale L^2 for every fixed d, which we achieve by a suitable
induction over the dimension together with an estimate for adjacent
transpositions. The lower bound is obtained with a version of Wilson's argument
for the one-dimensional case.Comment: 22 page
Relaxation time of anisotropic simple exclusion processes and quantum Heisenberg models
Motivated by an exact mapping between anisotropic half integer spin quantum
Heisenberg models and asymmetric diffusions on the lattice, we consider an
anisotropic simple exclusion process with particles in a rectangle of
\bbZ^2. Every particle at row tries to jump to an arbitrary empty site at
row with rate , where is a measure of the
drift driving the particles towards the bottom of the rectangle. We prove that
the spectral gap of the generator is uniformly positive in and in the size
of the rectangle. The proof is inspired by a recent interesting technique
envisaged by E. Carlen, M.C. Carvalho and M. Loss to analyze the Kac model for
the non linear Boltzmann equation. We then apply the result to prove precise
upper and lower bounds on the energy gap for the spin--S, {\rm S}\in
\frac12\bbN, XXZ chain and for the 111 interface of the spin--S XXZ Heisenberg
model, thus generalizing previous results valid only for spin .Comment: 27 page
Entropy dissipation estimates in a Zero-Range dynamics
We study the exponential decay of relative entropy functionals for zero-range
processes on the complete graph. For the standard model with rates increasing
at infinity we prove entropy dissipation estimates, uniformly over the number
of particles and the number of vertices
Isoperimetric inequalities and mixing time for a random walk on a random point process
We consider the random walk on a simple point process on ,
, whose jump rates decay exponentially in the -power of jump
length. The case corresponds to the phonon-induced variable-range
hopping in disordered solids in the regime of strong Anderson localization.
Under mild assumptions on the point process, we show, for ,
that the random walk confined to a cubic box of side has a.s. Cheeger
constant of order at least and mixing time of order . For the
Poisson point process, we prove that at , there is a transition from
diffusive to subdiffusive behavior of the mixing time.Comment: Published in at http://dx.doi.org/10.1214/07-AAP442 the Annals of
Applied Probability (http://www.imstat.org/aap/) by the Institute of
Mathematical Statistics (http://www.imstat.org
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