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

## TL;DR

This paper proves the Einstein relation for a biased random walk on a one-dimensional percolation cluster, showing the speed's differentiability at zero bias and connecting it to the walk's diffusivity.

## Contribution

It establishes the Einstein relation for the model, demonstrating the link between the derivative of speed at zero bias and the diffusivity, extending previous results.

## Key findings

- Speed is differentiable at zero bias
- Einstein relation holds for the model
- Unbiased walk diffusivity matches speed derivative

## 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.