An invariance principle for ergodic scale-free random environments

does the same with M and M ′ reversed.Note that the integrand is equal to e -r for each r>0 such an invariance principle for ergodic scale-free random environments 305 that no such homeomorphism exists.When we speak of random embedded lattices, we implicitly assume that their laws are defined w.r.t. the Borel σ -algebra associated with the topology defined by (1.1).It is a classical problem to determine whether a random walk on a random embedded lattice converges to Brownian motion (or to some other limiting process).There is a vast literature on this question, falling under the heading of random walk in random environment.See, e.g., [KLO], [BF], [Bi], [Ku] for recent surveys.Within this literature, the term random conductance model is used to describe walks which choose an edge to traverse (starting from the current vertex) with probability proportional to the conductance assigned to that edge.These are the types of walks we consider in this paper.Many existing results show the convergence (in the quenched sense) of random walk to Brownian motion for random embedded lattices that are translation invariant (stationary in law with respect to spatial translations), i.e., M-z d =M for each z∈C or for each z∈Z 2 , in the discrete case.Examples of such lattices include periodic lattices such as Z 2 with conductances assigned in a stationary way; the infinite cluster of supercritical percolation on such a lattice [BB]; and various graph structures on stationary point processes in C. Much of this work is based on the "environment seen from the particle" approach initiated in [KV], [DFGW].Brownian motion convergence results for random walks in random environments are often called sinvariance principles.