Biased random walk conditioned on survival among Bernoulli obstacles: subcritical phase

Abstract

We consider a discrete time biased random walk conditioned to avoid Bernoulli obstacles on ${\mathbb{Z}}^d$ ($d\ge 2$) up to time $N$. This model is known to undergo a phase transition: for a large bias, the walk is ballistic whereas for a small bias, it is sub-ballistic. We prove that in the sub-ballistic phase, the random walk is contained in a ball of radius $O(N^{1/(d+2)})$, which is the same scale as for the unbiased case. As an intermediate step, we also prove large deviation principles for the endpoint distribution for the unbiased random walk at scales between $N^{1/(d+2)}$ and $o(N^{d/(d+2)})$. These results improve and complement earlier work by Sznitman [Ann. Sci. Ecole Norm. Sup. (4), 28(3):345--370, 371--390, 1995].

arXiv comment34 pages, 3 figures. To appear in Commun. Math. Phys

References in corpus (7)

• Faber-Krahn inequalities in sharp quantitative form
• The Statistical Mechanics of Stretched Polymers
• Self-Attractive Random Walks: The Case of Critical Drifts
• Crossing velocities for an annealed random walk in a random potential
• Geometry of the random walk range conditioned on survival among Bernoulli obstacles
• The random walk penalised by its range in dimensions $d\ge 3$
• Localization for random walks among random obstacles in a single Euclidean ball

Cited by in corpus (1)

• Geometry of the random walk range conditioned on survival among Bernoulli obstacles