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Random walks in random sceneries (RWRS) are simple examples of stochasticprocesses in disordered media. They were introduced at the end of the 70's byKesten-Spitzer and Borodin, motivated by the construction of new self-similarprocesses with stationary increments. Two sources of randomness enter in theirdefinition: a random field $\xi = (\xi_x)_{x \in \Z^d}$ of i.i.d.\ randomvariables, 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 thetrajectory of the random walk, i.e., $Z_n := \sum_{k=1}^n \xi_{S_k}$. The lawof $Z$ under the joint law of $\xi$ and $S$ is called "annealed", and theconditional law given $\xi$ is called "quenched". Recently, central limittheorems under the quenched law were proved for $Z$ by the first two authorsfor a class of transient random walks including walks with finite variance indimension $d \ge 3$. In this paper we extend their results to dimension $d=2$.

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