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

摘要

Let {S-k, k >= 0) be a symmetric random walk on Z(d), and(eta (x), x is an element of Z(d)) an independent random field of centered i.i.d. random variables with tail decay P(eta(x) > t) approximate to exp(-t(alpha)). We consider a random walk in random scenery, that is Xn = eta (S-0) + - - - + eta (S-n). We present asymptotics for the probability, over both randomness, that {X-n > n(beta)) for beta > 1/2 and alpha > 1. To obtain such asymptotics, we establish large deviations estimates for the self-intersection local times process Sigma(x)l(n)(2)(x), where l(n) (x) is the number of visits of site x up to time n.