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

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