Annealed deviations of random walk in random scenery

Abstract

Let $(Z_n{)}_{n\in \mathbb{N}}$ be a $d$-dimensional {\it random walk in random scenery}, i.e., $Z_n={\sum }_{k=0}^{n-1}Y(S_k)$ with $(S_k{)}_{k\in {\mathbb{N}}_0}$ 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 steps have mean zero and finite variance. We identify the speed and the rate of the logarithmic decay of $\P (\frac{1}{n}{Z}_n>b_n)$ for various choices of sequences $(b_n{)}_n$ in $[1,\infty )$. Depending on $(b_n{)}_n$ and the upper tails of the scenery, we identify different regimes for the speed of decay and different variational formulas for the rate functions. In contrast to recent work \cite{AC02} by A. Asselah and F. Castell, we consider sceneries {\it unbounded} to infinity. It turns out that there are interesting connections to large deviation properties of self-intersections of the walk, which have been studied recently by X. Chen \cite{C03}.

32 pages, revised

References in corpus (1)

• A note on random walk in random scenery