# Regularity of the speed of biased random walk in a one-dimensional percolation model

## TL;DR

This paper studies the speed of biased random walks on a percolation cluster in a ladder graph, proving continuity and differentiability of the speed with respect to bias and establishing a central limit theorem in certain regimes.

## Contribution

It demonstrates the regularity properties of the walk's speed and characterizes its derivative, extending understanding of phase transition effects in this model.

## Key findings

- Speed is continuous in bias parameter below critical value.
- Speed is differentiable in bias parameter below half the critical value.
- Central limit theorem holds for bias below half the critical value, but not above.

## Abstract

We consider biased random walks on the infinite cluster of a conditional bond percolation model on the infinite ladder graph. Axelsson-Fisk and H\"aggstr\"om established for this model a phase transition for the asymptotic linear speed $\overline{\mathrm{v}}$ of the walk. Namely, there exists some critical value $\lambda_{\mathrm{c}}>0$ such that $\overline{\mathrm{v}}>0$ if $\lambda\in (0,\lambda_{\mathrm{c}})$ and $\overline{\mathrm{v}}=0$ if $\lambda>\lambda_{\mathrm{c}}$. We show that the speed $\overline{\mathrm{v}}$ is continuous in $\lambda$ on the interval $(0,\lambda_{\mathrm{c}})$ and differentiable on $(0,\lambda_{\mathrm{c}}/2)$. Moreover, we characterize the derivative as a covariance. For the proof of the differentiability of $\overline{\mathrm{v}}$ on $(0,\lambda_{\mathrm{c}}/2)$, we require and prove a central limit theorem for the biased random walk. Additionally, we prove that the central limit theorem fails to hold for $\lambda \geq \lambda_{\mathrm{c}}/2$.

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Source: https://tomesphere.com/paper/1705.00671