Random walk in random scenery and self-intersection local times in dimensions d ≥ 5

Abstract

Let {S k , k ≥ 0} be a symmetric random walk on $${\mathbb Z}^d$$ , and $$\{\eta(x), x\in {\mathbb Z^d\}}$$ an independent random field of centered i.i.d. random variables with tail decay $$P(\eta(x)> t)\approx\exp(-t^{\alpha})$$ . We consider a random walk in random scenery, that is $$X_n=\eta(S_0)+\dots+\eta(S_n)$$ . We present asymptotics for the probability, over both randomness, that {X n > n β} for β > 1/2 and α > 1. To obtain such asymptotics, we establish large deviations estimates for the self-intersection local times process $$\sum_x l_n^2(x)$$ , where l n (x) is the number of visits of site x up to time n.