A remark concerning random walks with random potentials

We consider random walks where each path is equipped with a random weight which is stationary and independent in space and time. We show that under some assumptions the arising probability distributions are in a sense uniformly absolutely continuous with respect to the usual probability distribution for symmetric random walks. We consider random walks on the d-dimensional lattice Z with each path having a random statistical weight. Paths starting at (x, k) and ending at (y, n) will be denoted by ω x,k , i.e. ω y,n x,k = {ω(t) ∈ Z, k ≤ t ≤ n , ω(k) = x, ω(n) = y, ‖ω(t + 1) − ω(t)‖ = 1}. To define a random weight introduce a sequence of iid rv F = {F (x, t)}, x ∈ Z, t ∈ Z. Without any loss of generality we may assume that the F (x, t) are given for all x ∈ Z, t ∈ Z. The space of all possible realizations of F is denoted by Φ. The measure corresponding to F is denoted by Q, the expectation with respect to Q is denoted by M . We do not use any special notation for the natural σ-algebra in Φ. Our main assumption concerning the distribution of F (x, t) is M exp(2F (x, t)) <∞. The natural group of space-time translations acting in Φ is denoted by {T x,t}. It preserves the measure Q. We shall consider the statistical weight of ω x,k equal to π(ω x,k ) = exp { n ∑ t=k F (t, ω(t)) } 1 (2d)n−k . Introduce partition functions Z x,k = ∑ ω x,k π(ω x,k ), Z n x,k = ∑