Abstract
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 $\xi = (\xi(x))_{x \in \mathbb{Z}^d}$ of i.i.d. random variables, which is called the random scenery, and a random walk $S = (S_n)_{n \in \mathbb{N}}$ evolving in $\mathbb{Z}^d$, independent of the scenery. The RWRS $Z = (Z_n)_{n \in \mathbb{N}}$ is then defined as the accumulated scenery along the trajectory of the random walk, i.e., $Z_n := \sum_{k=1}^n \xi(S_k)$. The law of $Z$ under the joint law of $\xi$ and $S$ is called annealed'', and the conditional law given $\xi$ 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 $d \ge 3$. In this paper we extend their results to dimension $d=2$.
Highlights
Let d ≥ 1 and (ξ(x))x∈Zd be a collection of independent and identically distributed (i.i.d.) real random variables, further referred to as scenery, and (Sn)n≥0 a random walk evolving in Zd, independent of the scenery
The random walk in random scenery (RWRS) is the process obtained by adding up the values of the scenery seen by the random walk along its trajectory, that is, Zn =
RWRS appears naturally in a variety of contexts, for instance (i) in the energy function of statistical mechanics models of polymers interacting with a random medium, (ii) in Bouchaud’s trap model via the clock process, see [2], (iii) in the study of random walks in randomly oriented lattices, as in [8, 11]
Summary
Let d ≥ 1 and (ξ(x))x∈Zd be a collection of independent and identically distributed (i.i.d.) real random variables, further referred to as scenery, and (Sn)n≥0 a random walk evolving in Zd, independent of the scenery. The case of a transient random walk (i.e., a < d) has been treated in [10] (see [29, 23, 7]): rescaling by n1/b one obtains as limit a stable process of index b. The first two authors√proved in [21] that a quenched functional central limit theorem (with the usual n-scaling and Gaussian law in the limit) holds for a class of transient random walks. The aim of this paper is to prove the following quenched functional central limit theorem. Sn n n converges in distribution to a random variable with characteristic function given by t → exp(−a|t|), a > 0
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