Some Remarks on the Large Deviation of the Visited Sites of Simple Random Walk in Random Scenery

Parkpoom Phetpradap

Abstract

For each $z \in \mathbb{Z}^d$, we define random scenery on the integer lattice $\mathbb{Z}^d$ as $\{ \xi_z: z \in \mathbb{Z}^d\}$ where each $\xi_z$ are identical and independent random variables with finite mean and variance. For a simple symmetric random walk on $\mathbb{Z}^d$ in dimension $d \geq 3$, we focus on $X_n:= \sum_{z \in V_n} \xi_z $, where $V_n$ is the lattice visited by the walk by time $n$. We investigate that $X_n$ satisfies large deviation principle with explicitly given rate functions. The expectation and variance of $X_n$ can also be calculated. This is an extended result from the large deviation result on the number of sites visited by a simple random walk.