Some remarks on the large deviation of the visited sites of simple random walk in random scenery

Abstract

© 2018 by the Mathematical Association of Thailand. All rights reserved. For each z ∈ Zd, we define random scenery on the integer lattice Zdas (Formula presented) where each ξzare identical and independent random variables with finite mean and variance. For a ∑simple symmetric random walk on Zdin dimension d ≥ 3, we focus on (Formula presented), where Vnis the lattice visited by the walk by time n. We investigate that Xnnsatisfies large deviation principle with explicitly given rate functions. The expectation and variance of Xncan also be calculated. This is an extended result from the large deviation result on the number of sites visited by a simple random walk.