The Random Conductance Model with Cauchy Tails1 By

0. Introduction. In this paper we will establish the convergence to Brownian motion of a random walk in a symmetric random environment in a critical case that has not been covered by the papers [1, 3]. We begin by recalling the “random conductance model” (RCM). We consider the Euclidean lattice Z with d ≥ 2. Let Ed be the set of nonoriented nearest neighbour bonds, and, writing e = {x, y} ∈ Ed , let (μe, e ∈ Ed) be nonnegative i.i.d. r.v. on [1,∞) defined on a probability space ( ,P). We write μxy = μ{x,y} = μyx ; let μxy = 0 if x ∼ y, and set μx = ∑ y μxy . We consider two continuous time random walks on Z which jump from x to y ∼ x with probability μxy/μx . These are called in [1] the constant speed random walk (CSRW) and variable speed random walk (VSRW), and have generators