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

For each z E Z(d), we define random scenery on the integer lattice Zd as {z E Z(d)} where eachz are identical and independent random variables with finite mean and variance. For a simple symmetric random walk on Zd in dimension d > 3, we focus on X := -zEVz where 17Th 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 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.