TY - JOUR

T1 - Regularity of the Speed of Biased Random Walk in a One-Dimensional Percolation Model

AU - Gantert, Nina

AU - Meiners, Matthias

AU - Müller, Sebastian

N1 - Publisher Copyright:
© 2018, The Author(s).

PY - 2018/3/1

Y1 - 2018/3/1

N2 - We consider biased random walks on the infinite cluster of a conditional bond percolation model on the infinite ladder graph. Axelson-Fisk and Häggström established for this model a phase transition for the asymptotic linear speed v ¯ of the walk. Namely, there exists some critical value λc> 0 such that v ¯ > 0 if λ∈ (0 , λc) and v ¯ = 0 if λ≥ λc. We show that the speed v ¯ is continuous in λ on (0 , ∞) and differentiable on (0 , λc/ 2). Moreover, we characterize the derivative as a covariance. For the proof of the differentiability of v ¯ on (0 , λc/ 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 λ≥ λc/ 2.

AB - We consider biased random walks on the infinite cluster of a conditional bond percolation model on the infinite ladder graph. Axelson-Fisk and Häggström established for this model a phase transition for the asymptotic linear speed v ¯ of the walk. Namely, there exists some critical value λc> 0 such that v ¯ > 0 if λ∈ (0 , λc) and v ¯ = 0 if λ≥ λc. We show that the speed v ¯ is continuous in λ on (0 , ∞) and differentiable on (0 , λc/ 2). Moreover, we characterize the derivative as a covariance. For the proof of the differentiability of v ¯ on (0 , λc/ 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 λ≥ λc/ 2.

KW - Biased random walk

KW - Invariance principle

KW - Ladder graph

KW - Percolation

KW - Regularity of the speed

UR - http://www.scopus.com/inward/record.url?scp=85041902429&partnerID=8YFLogxK

U2 - 10.1007/s10955-018-1982-4

DO - 10.1007/s10955-018-1982-4

M3 - Article

AN - SCOPUS:85041902429

SN - 0022-4715

VL - 170

SP - 1123

EP - 1160

JO - Journal of Statistical Physics

JF - Journal of Statistical Physics

IS - 6

ER -