Annealed deviations of random walk in random scenery

Abstract

Let ( Z n ) n ∈ N be a d-dimensional random walk in random scenery, i.e., Z n = ∑ k = 0 n − 1 Y ( S k ) with ( S k ) k ∈ N 0 a random walk in Z d and ( Y ( z ) ) z ∈ Z d an i.i.d. scenery, independent of the walk. The walker's steps have mean zero and some finite exponential moments. We identify the speed and the rate of the logarithmic decay of P ( 1 n Z n > b n ) for various choices of sequences ( b n ) n in [ 1 , ∞ ) . Depending on ( b n ) n and the upper tails of the scenery, we identify different regimes for the speed of decay and different variational formulas for the rate functions. In contrast to recent work [A. Asselah, F. Castell, Large deviations for Brownian motion in a random scenery, Probab. Theory Related Fields 126 (2003) 497–527] by A. Asselah and F. Castell, we consider sceneries unbounded to infinity. It turns out that there are interesting connections to large deviation properties of self-intersections of the walk, which have been studied recently by X. Chen [X. Chen, Exponential asymptotics and law of the iterated logarithm for intersection local times of random walks, Ann. Probab. 32 (4) 2004].