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Annealed deviations of random walk in random scenery. (English) Zbl 1119.60083
Let $$(Z_n)_{n=1,2,\dots}$$ be a $$d$$-dimensional random walk in random scenery, i.e., $$Z_n=\sum_{k=0}^{n-1} Y(S_k)$$ with $$(S_k)_{k=0,1,\dots}$$ a random walk in $$\mathbb{Z}^d$$ and $$(Y(z))_{z \in \mathbb{Z}^d}$$ an i.i.d. scenery, independent of the walk. The walker’s step has mean zero and some finite exponential moments. The speed and the rate of $$\log P(n^{-1}Z_n >b_n)$$ are identified for various choices of sequences $$(b_n)_{n=1,2,\dots}$$. For example, the so-called very large deviations are defined by requirements $$1 \ll b_n\ll n^{1/q}$$ for some $$q>0$$. Depending on $$(b_n)_n$$ and the upper tails of the scenery, the authors identify different regimes for the speed of $$\log P(n^{-1}Z_n >b_n)$$ and different variational formulas for the rate functions. The considered problem has been previously addressed in the work of A. Asselah and F. Castell [Probab. Theory Relat. Fields 126, No. 4, 497–527 (2003; Zbl 1043.60018)]. The main novelty of the paper is the study of arbitrary sceneries unbounded to $$+\infty$$ and general scale functions $$b_n \gg 1$$ in the discrete setting.
The results of the article have interesting connections to large deviation properties of self-intersections of the walk, which have been studied by X. Chen [Ann. Probab. 32, No. 4, 3248–3300 (2004; Zbl 1067.60071)].

##### MSC:
 60K37 Processes in random environments 60F10 Large deviations 60J55 Local time and additive functionals
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