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

- 2013

#### 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 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.