134 research outputs found
On the scaling of the chemical distance in long-range percolation models
We consider the (unoriented) long-range percolation on Z^d in dimensions
d\ge1, where distinct sites x,y\in Z^d get connected with probability
p_{xy}\in[0,1]. Assuming p_{xy}=|x-y|^{-s+o(1)} as |x-y|\to\infty, where s>0
and |\cdot| is a norm distance on Z^d, and supposing that the resulting random
graph contains an infinite connected component C_{\infty}, we let D(x,y) be the
graph distance between x and y measured on C_{\infty}. Our main result is that,
for s\in(d,2d), D(x,y)=(\log|x-y|)^{\Delta+o(1)},\qquad x,y\in C_{\infty},
|x-y|\to\infty, where \Delta^{-1} is the binary logarithm of 2d/s and o(1) is a
quantity tending to zero in probability as |x-y|\to\infty. Besides its interest
for general percolation theory, this result sheds some light on a question that
has recently surfaced in the context of ``small-world'' phenomena. As part of
the proof we also establish tight bounds on the probability that the largest
connected component in a finite box contains a positive fraction of all sites
in the box.Comment: Published at http://dx.doi.org/10.1214/009117904000000577 in the
Annals of Probability (http://www.imstat.org/aop/) by the Institute of
Mathematical Statistics (http://www.imstat.org
Scaling limit for a class of gradient fields with nonconvex potentials
We consider gradient fields whose law takes the
Gibbs--Boltzmann form , where the
sum runs over nearest neighbors. We assume that the potential admits the
representation V(\eta):=-\log\int\varrho({d}\kappa)\exp\biggl[-{1/2}\kappa\et
a^2\biggr], where is a positive measure with compact support in
. Hence, the potential is symmetric, but nonconvex in general.
While for strictly convex 's, the translation-invariant, ergodic gradient
Gibbs measures are completely characterized by their tilt, a nonconvex
potential as above may lead to several ergodic gradient Gibbs measures with
zero tilt. Still, every ergodic, zero-tilt gradient Gibbs measure for the
potential above scales to a Gaussian free field.Comment: Published in at http://dx.doi.org/10.1214/10-AOP548 the Annals of
Probability (http://www.imstat.org/aop/) by the Institute of Mathematical
Statistics (http://www.imstat.org
Eigenvalue order statistics for random Schr\"odinger operators with doubly-exponential tails
We consider random Schr\"odinger operators of the form , where
is the lattice Laplacian on and is an i.i.d.
random field, and study the extreme order statistics of the eigenvalues for
this operator restricted to large but finite subsets of . We show
that for with a doubly-exponential type of upper tail, the upper extreme
order statistics of the eigenvalues falls into the Gumbel max-order class. The
corresponding eigenfunctions are exponentially localized in regions where
takes large, and properly arranged, values. A new and self-contained argument
is thus provided for Anderson localization at the spectral edge which permits a
rather explicit description of the shape of the potential and the
eigenfunctions. Our study serves as an input into the analysis of an associated
parabolic Anderson problem.Comment: 36 page
Forbidden gap argument for phase transitions proved by means of chessboard estimates
Chessboard estimates are one of the standard tools for proving phase
coexistence in spin systems of physical interest. In this note we show that the
method not only produces a point in the phase diagram where more than one Gibbs
states coexist, but that it can also be used to rule out the existence of
shift-ergodic states that differ significantly from those proved to exist.
For models depending on a parameter (say, the temperature), this shows that
the values of the conjugate thermodynamic quantity (the energy) inside the
"transitional gap" are forbidden in all shift-ergodic Gibbs states. We point
out several models where our result provides useful additional information
concerning the set of possible thermodynamic equilibria.Comment: 26 page
Gibbs measures on permutations over one-dimensional discrete point sets
We consider Gibbs distributions on permutations of a locally finite infinite
set , where a permutation of is assigned
(formal) energy . This is motivated by Feynman's
path representation of the quantum Bose gas; the choice and
is of principal interest. Under suitable regularity
conditions on the set and the potential , we establish existence and a
full classification of the infinite-volume Gibbs measures for this problem,
including a result on the number of infinite cycles of typical permutations.
Unlike earlier results, our conclusions are not limited to small densities
and/or high temperatures.Comment: Published in at http://dx.doi.org/10.1214/14-AAP1013 the Annals of
Applied Probability (http://www.imstat.org/aap/) by the Institute of
Mathematical Statistics (http://www.imstat.org
Quenched invariance principle for simple random walk on percolation clusters
We consider the simple random walk on the (unique) infinite cluster of
super-critical bond percolation in with . We prove that, for
almost every percolation configuration, the path distribution of the walk
converges weakly to that of non-degenerate, isotropic Brownian motion. Our
analysis is based on the consideration of a harmonic deformation of the
infinite cluster on which the random walk becomes a square-integrable
martingale. The size of the deformation, expressed by the so called corrector,
is estimated by means of ergodicity arguments.Comment: 38 pages (PTRF format) 4 figures. Version to appear in PTR
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