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Random walk in random scenery and self-intersection local times in dimensions \(d \geq 5\). (English) Zbl 1116.60057

Summary: Let \(\{S_{k}\), \(k \geq 0\}\) 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 \(X_n = \eta (S_0) +\cdots+\eta(S_n)\). 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 deviation 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 \(n\).

MSC:

60K37 Processes in random environments
60F10 Large deviations
60J55 Local time and additive functionals
60G50 Sums of independent random variables; random walks
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