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

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 \emph{random scenery}, and a random walk $S = (S_n){n \in \N}$evolvingin$\Z^d$,independentfromthescenery\mathrm{.}TheRWRS$Z = (Z_n){n \in \N}$ 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 $d=2$.