193 research outputs found
Matching with shift for one-dimensional Gibbs measures
We consider matching with shifts for Gibbsian sequences. We prove that the
maximal overlap behaves as , where is explicitly identified in
terms of the thermodynamic quantities (pressure) of the underlying potential.
Our approach is based on the analysis of the first and second moment of the
number of overlaps of a given size. We treat both the case of equal sequences
(and nonzero shifts) and independent sequences.Comment: Published in at http://dx.doi.org/10.1214/08-AAP588 the Annals of
Applied Probability (http://www.imstat.org/aap/) by the Institute of
Mathematical Statistics (http://www.imstat.org
A concentration inequality for interval maps with an indifferent fixed point
For a map of the unit interval with an indifferent fixed point, we prove an
upper bound for the variance of all observables of variables
which are componentwise Lipschitz. The proof is based on
coupling and decay of correlation properties of the map. We then give various
applications of this inequality to the almost-sure central limit theorem, the
kernel density estimation, the empirical measure and the periodogram.Comment: 26 pages, submitte
On the Hydrodynamic Equilibrium of a Rod in a Lattice Fluid
We model the behavior of a big (Brazil) nut in a medium of smaller nuts with
a stochastic asymmetric simple exclusion dynamics of a polymer-monomer lattice
system. The polymer or `rod' can move up or down in an external negative field,
occupying N horizontal lattice sites where the monomers cannot enter. The
monomers (at most one per site) or `fluid particles' are moving symmetrically
in the horizontal plane and asymmetrically in the vertical direction, also with
a negative field. For a fixed position of the rod, this lattice fluid is in
equilibrium with a vertical height profile reversible for the monomers' motion.
Upon `shaking' (speeding up the monomers) the motion of the `rod' dynamically
decouples from that of the monomers resulting in a reversible random walk for
the rod around an average height proportional to log N.Comment: 19 pages, 2 figure
Limiting shapes for deterministic centrally seeded growth models
We study the rotor router model and two deterministic sandpile models. For
the rotor router model in , Levine and Peres proved that the
limiting shape of the growth cluster is a sphere. For the other two models,
only bounds in dimension 2 are known. A unified approach for these models with
a new parameter (the initial number of particles at each site), allows to
prove a number of new limiting shape results in any dimension .
For the rotor router model, the limiting shape is a sphere for all values of
. For one of the sandpile models, and (the maximal value), the
limiting shape is a cube. For both sandpile models, the limiting shape is a
sphere in the limit . Finally, we prove that the rotor router
shape contains a diamond.Comment: 18 pages, 3 figures, some errors corrected and more explanation
added, to appear in Journal of Statistical Physic
Concentration inequalities for random fields via coupling
We present a new and simple approach to concentration inequalities for
functions around their expectation with respect to non-product measures, i.e.,
for dependent random variables. Our method is based on coupling ideas and does
not use information inequalities. When one has a uniform control on the
coupling, this leads to exponential concentration inequalities. When such a
uniform control is no more possible, this leads to polynomial or
stretched-exponential concentration inequalities. Our abstract results apply to
Gibbs random fields, in particular to the low-temperature Ising model which is
a concrete example of non-uniformity of the coupling.Comment: New corrected version; 22 pages; 1 figure; New result added:
stretched-exponential inequalit
A probabilistic approach to Zhang's sandpile model
The current literature on sandpile models mainly deals with the abelian
sandpile model (ASM) and its variants. We treat a less known - but equally
interesting - model, namely Zhang's sandpile. This model differs in two aspects
from the ASM. First, additions are not discrete, but random amounts with a
uniform distribution on an interval . Second, if a site topples - which
happens if the amount at that site is larger than a threshold value
(which is a model parameter), then it divides its entire content in equal
amounts among its neighbors. Zhang conjectured that in the infinite volume
limit, this model tends to behave like the ASM in the sense that the stationary
measure for the system in large volumes tends to be peaked narrowly around a
finite set. This belief is supported by simulations, but so far not by
analytical investigations.
We study the stationary distribution of this model in one dimension, for
several values of and . When there is only one site, exact computations
are possible. Our main result concerns the limit as the number of sites tends
to infinity, in the one-dimensional case. We find that the stationary
distribution, in the case , indeed tends to that of the ASM (up
to a scaling factor), in agreement with Zhang's conjecture. For the case ,
we provide strong evidence that the stationary expectation tends to
.Comment: 47 pages, 3 figure
Short-time Gibbsianness for Infinite-dimensional Diffusions with Space-Time Interaction
We consider a class of infinite-dimensional diffusions where the interaction
between the components is both spatial and temporal. We start the system from a
Gibbs measure with finite-range uniformly bounded interaction. Under suitable
conditions on the drift, we prove that there exists such that the
distribution at time is a Gibbs measure with absolutely summable
interaction. The main tool is a cluster expansion of both the initial
interaction and certain time-reversed Girsanov factors coming from the
dynamics
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