A Quenched Functional Central Limit Theorem for Planar Random Walks in Random Sceneries

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 $\xi = (\xi_x){x \in \Z^d}$ of i.i.d.\ random variables, which is called the \emph{random scenery}, and a random walk $S = (S_n){n \in \N}$ evolving in $\Z^d$, independent from the scenery. The RWRS $Z = (Z_n){n \in \N}$ 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, 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 $d=2$.