Quenched central limit theorems for random walks in random scenery

Random walks in random scenery are processes defined by Zn:=∑k=1nωSk where S:=(Sk,k≥0)S:=(Sk,k≥0) is a random walk evolving in ZdZd and ω:=(ωx,x∈Zd)ω:=(ωx,x∈Zd) is a sequence of i.i.d. real random variables. Under suitable assumptions on the random walk SS and the random scenery ωω, almost surely with respect to ωω, the correctly renormalized sequence (Zn)n≥1(Zn)n≥1 is proved to converge in distribution to a centered Gaussian law with explicit variance.