In this paper we study a random walk in a one-dimensional dynamic random environment consisting of a collection of independent particles performing simple symmetric random walks in a Poisson equilibrium with density ρ ∈ (0, ∞). At each step the random walk performs a nearest-neighbour jump, moving to the right with probability p • when it is on a vacant site and probability p • when it is on an occupied site. Assuming that p • ∈ (0, 1) and p • = 1 2 , we show that the position of the random walk satisfies a strong law of large numbers, a functional central limit theorem and a large deviation bound, provided ρ is large enough. The proof is based on the construction of a renewal structure together with a multiscale renormalisation argument.MSC 2010. Primary 60F15, 60K35, 60K37; Secondary 82B41, 82C22, 82C44.

ABSTRACT. We study the non-negative solution u = u(x, t) to the Cauchy problem for the parabolic equationHere ∆ is the discrete Laplacian on Z d and ξ = (ξ(z)) z∈Z d is an i.i.d. random field with doubly-exponential upper tails. We prove that, for large t and with large probability, most of the total mass U(t) := ∑ x u(x, t) of the solution resides in a bounded neighborhood of a site Z t that achieves an optimal compromise between the local Dirichlet eigenvalue of the Anderson Hamiltonian ∆ + ξ and the distance to the origin. The processes t → Z t and t → 1 t log U(t) are shown to converge in distribution under suitable scaling of space and time. Aging results for Z t , as well as for the solution to the parabolic problem, are also established. The proof uses the characterization of eigenvalue order statistics for ∆ + ξ in large sets recently proved by the first two authors.

We study the evolution of a random walker on a conservative dynamic random environment composed of independent particles performing simple symmetric random walks, generalizing results of [16] to higher dimensions and more general transition kernels without the assumption of uniform ellipticity or nearest-neighbour jumps. Specifically, we obtain a strong law of large numbers, a functional central limit theorem and large deviation estimates for the position of the random walker under the annealed law in a high density regime. The main obstacle is the intrinsic lack of monotonicity in higher-dimensional, non-nearest neighbour settings. Here we develop more general renormalization and renewal schemes that allow us to overcome this issue. As a second application of our methods, we provide an alternative proof of the ballistic behaviour of the front of (the discrete-time version of) the infection model introduced in [23].

Random walks in random sceneries (RWRS) are simple examples of stochastic processes in disordered media. They were introduced at the end of the 70's by Kesten-Spitzer and Borodin, motivated by the construction of new self-similar processes with stationary increments. Two sources of randomness enter in their definition: a random field ξ = (ξ(x)) x∈Z d of i.i.d. random variables, which is called the random scenery, and a random walk S = (Sn) n∈N evolving in Z d , independent of the scenery. The RWRS Z = (Zn) n∈N is then defined as the accumulated scenery along the trajectory of the random walk, i.e., Zn := n k=1 ξ(S k ). The law of Z under the joint law of ξ and S is called "annealed", and the conditional law given ξ is called "quenched". Recently, functional central limit theorems under the quenched law were proved for Z by the first two authors for a class of transient random walks including walks with finite variance in dimension d ≥ 3. In this paper we extend their results to dimension d = 2.

We 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 [9] and Berger and Zeitouni in [2] under the assumption of large finite moments for the regeneration time. In this paper, with the extra (T )γ condition of Sznitman we reduce the moment condition to E(τ 2 (ln τ ) 1+m ) < +∞ for m > 1 + 1/γ, which allows the inclusion of new non-uniformly elliptic examples such as Dirichlet random environments.