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 < γ - < γ + < + ∞ , 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.