37 research outputs found
The Point of View of the Particle on the Law of Large Numbers for Random Walks in a Mixing Random Environment
The point of view of the particle is an approach that has proven very
powerful in the study of many models of random motions in random media. We
provide a new use of this approach to prove the law of large numbers in the
case of one or higher-dimensional, finite range, transient random walks in
mixing random environments. One of the advantages of this method over what has
been used so far is that it is not restricted to i.i.d. environments.Comment: 22 pages. To appear in the Annals of Probabilit
Almost sure functional central limit theorem for non-nestling random walk in random environment
We consider a non-nestling random walk in a product random environment. We
assume an exponential moment for the step of the walk, uniformly in the
environment. We prove an invariance principle (functional central limit
theorem) under almost every environment for the centered and diffusively scaled
walk. The main point behind the invariance principle is that the quenched mean
of the walk behaves subdiffusively.Comment: 54 pages. Small edits in tex
Almost sure functional central limit theorem for ballistic random walk in random environment
We consider a multidimensional random walk in a product random environment
with bounded steps, transience in some spatial direction, and high enough
moments on the regeneration time. We prove an invariance principle, or
functional central limit theorem, under almost every environment for the
diffusively scaled centered walk. The main point behind the invariance
principle is that the quenched mean of the walk behaves subdiffusively.Comment: Accepted to the Annales de l'Institut Henri Poincar
Almost Sure Invariance Principle for Continuous-Space Random Walk in Dynamic Random Environment
We consider a random walk on in a polynomially mixing random
environment that is refreshed at each time step. We use a martingale approach
to give a necessary and sufficient condition for the almost-sure functional
central limit theorem to hold.Comment: minor typos fixe
Process-level quenched large deviations for random walk in random environment
We consider a bounded step size random walk in an ergodic random environment
with some ellipticity, on an integer lattice of arbitrary dimension. We prove a
level 3 large deviation principle, under almost every environment, with rate
function related to a relative entropy.Comment: Proof of (6.2) corrected. Lemma A.2 replace
Quenched Point-to-Point Free Energy for Random Walks in Random Potentials
We consider a random walk in a random potential on a square lattice of
arbitrary dimension. The potential is a function of an ergodic environment and
some steps of the walk. The potential can be unbounded, but it is subject to a
moment assumption whose strictness is tied to the mixing of the environment,
the best case being the i.i.d. environment. We prove that the infinite volume
quenched point-to-point free energy exists and has a variational formula in
terms of an entropy. We establish regularity properties of the point-to-point
free energy, as a function of the potential and as a function on the convex
hull of the admissible steps of the walk, and link it to the infinite volume
free energy and quenched large deviations of the endpoint of the walk. One
corollary is a quenched large deviation principle for random walk in an ergodic
random environment, with a continuous rate function.Comment: 39 pages, 3 figures, minor typos fixe
Quenched invariance principle for multidimensional ballistic random walk in a random environment with a forbidden direction
We consider a ballistic random walk in an i.i.d. random environment that does
not allow retreating in a certain fixed direction. We prove an invariance
principle (functional central limit theorem) under almost every fixed
environment. The assumptions are nonnestling, at least two spatial dimensions,
and a moment for the step of the walk uniformly in the
environment. The main point behind the invariance principle is that the
quenched mean of the walk behaves subdiffusively.Comment: Published at http://dx.doi.org/10.1214/009117906000000610 in the
Annals of Probability (http://www.imstat.org/aop/) by the Institute of
Mathematical Statistics (http://www.imstat.org