166 research outputs found
Transience, Recurrence and Critical Behavior for Long-Range Percolation
We study the behavior of the random walk on the infinite cluster of
independent long range percolation in dimensions , where and a
re connected with probability . We show that when
the walk is transient, and when , the walk is recurrent. The
proof of transience is based on a renormalization argument. As a corollary of
this renormalization argument, we get that for every dimension , if
, then critical percolation has no infinite clusters. This result is
extended to the free random cluster model. A second corollary is that when
and we can erase all long enough bonds and still have an
infinite cluster. The proof of recurrence in two dimensions is based on general
stability results for recurrence in random electrical networks. In particular,
we show that i.i.d. conductances on a recurrent graph of bounded degree yield a
recurrent electrical network.Comment: Mario Wuetrich pointed out a gap in one of the proofs in this (more
than 10 years) old paper. Here is the corrected version. 43 pages, no figure
Limiting velocity of high-dimensional random walk in random environment
We show that random walk in uniformly elliptic i.i.d. environment in
dimension has at most one non zero limiting velocity. In particular
this proves a law of large numbers in the distributionally symmetric case and
establishes connections between different conjectures.Comment: Published in at http://dx.doi.org/10.1214/07-AOP338 the Annals of
Probability (http://www.imstat.org/aop/) by the Institute of Mathematical
Statistics (http://www.imstat.org
A quenched invariance principle for certain ballistic random walks in i.i.d. environments
We prove that every random walk in i.i.d. environment in dimension greater
than or equal to 2 that has an almost sure positive speed in a certain
direction, an annealed invariance principle and some mild integrability
condition for regeneration times also satisfies a quenched invariance
principle. The argument is based on intersection estimates and a theorem of
Bolthausen and Sznitman.Comment: This version includes an extension of the results to cover also
dimensions 2,3, and also corrects several minor innacuracies. The previous
version included a correction of a minor error in (3.21) (used for d=4); The
correction pushed the assumption on moments of regeneration times to >
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
On the speed of Random Walks among Random Conductances
We consider random walk among random conductances where the conductance
environment is shift invariant and ergodic. We study which moment conditions of
the conductances guarantee speed zero of the random walk. We show that if there
exists \alpha>1 such that E[log^\alpha({\omega}_e)]<\infty, then the random
walk has speed zero. On the other hand, for each \alpha>1 we provide examples
of random walks with non-zero speed and random walks for which the limiting
speed does not exist that have E[log^\alpha({\omega}_e)]<\infty.Comment: 22 pages, 4 picture
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