Einstein relation for random walk in a one-dimensional percolation model

Abstract

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.

References in corpus (4)

• Quenched invariance principle for simple random walk on percolation clusters
• Einstein relation for random walks in random environment
• Moments of random sums and Robbins' problem of optimal stopping
• Regularity of the speed of biased random walk in a one-dimensional percolation model

Cited by in corpus (2)

• The tail of the length of an excursion in a trap of random size
• Biased Random Walk on Spanning Trees of the Ladder Graph