Einstein relation for random walks in random environments

We consider a tracer particle performing a nearest neighbor random walk on Z(d) in dimension d >= 3 with random jump rates. This kind of a walk models the motion of a charged particle under a constant external electric field. We assume that the jump rates admit only two values 0 < y(-) < y(+) < + infinity, representing the lower and upper conductivities. We prove the existence of the mobility coefficient and that it equals to the diffusivity coefficient of the particle in zero external field. (c) 2005 Elsevier B.V. All rights reserved.