Law of large numbers for non-elliptic random walks in dynamic random environments

We prove a law of large numbers for a class of $\Z^d$-valued random walks in dynamic random environments, including non-elliptic examples. We assume for the random environment a mixing property called \emph{conditional cone-mixing} and that the random walk tends to stay inside wide enough space-time cones. The proof is based on a generalization of a regeneration scheme developed by Comets and Zeitouni for static random environments and adapted by Avena, den Hollander and Redig to dynamic random environments. A number of one-dimensional examples are given. In some cases, the sign of the speed can be determined.Comment: 36 pages, 4 figure

A Quenched Functional Central Limit Theorem for Random Walks in Random Environments under (T)γ(T)_\gamma

International audienceWe prove a quenched central limit theorem for random walks in i.i.d. weakly elliptic random environments in the ballistic regime. Such theorems have been proved recently by Rassoul-Agha and Seppäläinen in [10] and Berger and Zeitouni in [2] under the assumption of large finite moments for the regeneration time. In this paper, with the extra $(T)_{\gamma}$ condition of Sznitman we reduce the moment condition to ${\Bbb E}(\tau^2(\ln \tau)^{1+m})1+1/\gamma$, which allows the inclusion of new non-uniformly elliptic examples such as Dirichlet random environments

Fluctuations of the front in a stochastic combustion model

We consider an interacting particle system on the one dimensional lattice $\bf Z$ modeling combustion. The process depends on two integer parameters $2\le a<M<\infty$. Particles move independently as continuous time simple symmetric random walks except that 1. When a particle jumps to a site which has not been previously visited by any particle, it branches into $a$ particles; 2. When a particle jumps to a site with $M$ particles, it is annihilated. We start from a configuration where all sites to the left of the origin have been previously visited and study the law of large numbers and central limit theorem for $r_t$, the rightmost visited site at time $t$. The proofs are based on the construction of a renewal structure leading to a definition of regeneration times for which good tail estimates can be performed.Comment: 19 page

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 >