LAUSR

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 $$\uplambda > 0$$λ>0, then its asymptotic linear speed $$\overline{\mathrm {v}}$$v¯ is continuous in the variable $$\uplambda > 0$$λ>0 and differentiable for all sufficiently small $$\uplambda > 0$$λ>0. In the paper at hand, we complement this result by proving that $$\overline{\mathrm {v}}$$v¯ is differentiable at $$\uplambda = 0$$λ=0. Further, we show the Einstein relation for the model, i.e., that the derivative of the speed at $$\uplambda = 0$$λ=0 equals the diffusivity of the unbiased walk.