510 research outputs found
An almost sure invariance principle for random walks in a space-time random environment
We consider a discrete time random walk in a space-time i.i.d. random
environment. We use a martingale approach to show that the walk is diffusive in
almost every fixed environment. We improve on existing results by proving an
invariance principle and considering environments with an annealed drift.
We also state an a.s. invariance principle for random walks in general random
environments whose hypothesis requires a subdiffusive bound on the variance of
the quenched mean, under an ergodic invariant measure for the environment
chain
The quenched invariance principle for random walks in random environments admitting a bounded cycle representation
We derive a quenched invariance principle for random walks in random
environments whose transition probabilities are defined in terms of weighted
cycles of bounded length. To this end, we adapt the proof for random walks
among random conductances by Sidoravicius and Sznitman (Probab. Theory Related
Fields 129 (2004) 219--244) to the non-reversible setting.Comment: Published in at http://dx.doi.org/10.1214/07-AIHP122 the Annales de
l'Institut Henri Poincar\'e - Probabilit\'es et Statistiques
(http://www.imstat.org/aihp/) by the Institute of Mathematical Statistics
(http://www.imstat.org
Diffusive and Super-Diffusive Limits for Random Walks and Diffusions with Long Memory
We survey recent results of normal and anomalous diffusion of two types of
random motions with long memory in or . The first
class consists of random walks on in divergence-free random drift
field, modelling the motion of a particle suspended in time-stationary
incompressible turbulent flow. The second class consists of self-repelling
random diffusions, where the diffusing particle is pushed by the negative
gradient of its own occupation time measure towards regions less visited in the
past. We establish normal diffusion (with square-root-of-time scaling and
Gaussian limiting distribution) in three and more dimensions and typically
anomalously fast diffusion in low dimensions (typically, one and two). Results
are quoted from various papers published between 2012-2018, with some hints to
the main ideas of the proofs. No technical details are presented here.Comment: ICM-2018 Probability Section tal
Biased random walks on random graphs
These notes cover one of the topics programmed for the St Petersburg School
in Probability and Statistical Physics of June 2012.
The aim is to review recent mathematical developments in the field of random
walks in random environment. Our main focus will be on directionally transient
and reversible random walks on different types of underlying graph structures,
such as , trees and for .Comment: Survey based one of the topics programmed for the St Petersburg
School in Probability and Statistical Physics of June 2012. 64 pages, 16
figure
Quenched invariance principles for the random conductance model on a random graph with degenerate ergodic weights
We consider a stationary and ergodic random field
that is parameterized by the edge set of the Euclidean lattice ,
. The random variable , taking values in and
satisfying certain moment bounds, is thought of as the conductance of the edge
. Assuming that the set of edges with positive conductances give rise to a
unique infinite cluster , we prove a quenched
invariance principle for the continuous-time random walk among random
conductances under relatively mild conditions on the structure of the infinite
cluster. An essential ingredient of our proof is a new anchored relative
isoperimetric inequality.Comment: 22 page
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