Einstein Relation for Random Walk in a One-Dimensional Percolation Model

We consider random walks on the infinite cluster of a conditional bond percolation model on the infinite ladder graph. In a companion paper, we have shown that if the random walk is pulled to the right by a positive bias λ > 0 , then its asymptotic linear speed v ¯ is continuous in the variable λ > 0 and differentiable for all sufficiently small λ > 0. In the paper at hand, we complement this result by proving that v ¯ is differentiable at λ = 0. Further, we show the Einstein relation for the model, i.e., that the derivative of the speed at λ = 0 equals the diffusivity of the unbiased walk.