Regularity of the Speed of Biased Random Walk in a One-Dimensional Percolation Model

Nina Gantert (1) , Matthias Meiners (2) , Sebastian Müller (3)

1 Lehrstuhl fur Warscheinlichkeitstheorie
2 Universität Innsbruck [Innsbruck]
3 I2M - Institut de Mathématiques de Marseille

We consider biased random walks on the infinite cluster of a conditional bond percolation model on the infinite ladder graph. Axelson-Fisk and Haggstrom established for this model a phase transition for the asymptotic linear speed (v) over bar of the walk. Namely, there exists some critical value lambda(c) > 0 such that (v) over bar > 0 if lambda is an element of (0, lambda(c)) and (v) over bar = 0 if lambda >= lambda(c). We show that the speed (v) over bar is continuous in lambda on (0, infinity) and differentiable on (0, lambda(c)/2). Moreover, we characterize the derivative as a covariance. For the proof of the differentiability of (v) over bar on (0, lambda(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 >= lambda(c)/2.

Mots clés: Biased random walk; Regularity of the speed; Invariance principle; Ladder graph; Percolation

Domaines: Probabilités [math.PR]

Nina Gantert, Matthias Meiners, Sebastian Müller. Regularity of the Speed of Biased Random Walk in a One-Dimensional Percolation Model. 2017. ⟨hal-02051736⟩