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.

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 $\lambda \geq \lambda_{\mathrm{c}}/2$.