Random walks with unbounded jumps among random conductances I: Uniform quenched CLT

Abstract

We study a one-dimensional random walk among random conductances, with unbounded jumps. Assuming the ergodicity of the collection of conductances and a few other technical conditions (uniform ellipticity and polynomial bounds on the tails of the jumps) we prove a quenched \textit{uniform} invariance principle for the random walk. This means that the rescaled trajectory of length $n$ is (in a certain sense) close enough to the Brownian motion, uniformly with respect to the choice of the starting location in an interval of length $O(\sqrt{n})$ around the origin.

arXiv admin note: substantial text overlap with arXiv:1011.1196

References in corpus (2)

• Invariance principle for the random conductance model with unbounded conductances
• Random walks with unbounded jumps among random conductances II: Conditional quenched CLT

Cited by in corpus (2)

• Random walks with unbounded jumps among random conductances II: Conditional quenched CLT
• Law of large numbers for random walk with unbounded jumps and BDP with bounded jumps in random environment