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.
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Asselah, A., Castell, F. Random walk in random scenery and self-intersection local times in dimensions d ≥ 5. Probab. Theory Relat. Fields 138, 1–32 (2007). https://doi.org/10.1007/s00440-006-0014-5
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DOI: https://doi.org/10.1007/s00440-006-0014-5