Conditional and uniform quenched CLTs for one-dimensional random walks among random conductances

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{conditional} invariance principle for the random walk, under the condition that it remains positive until time $n$. As a corollary of this result, we study the effect of conditioning the random walk to exceed level $n$ before returning to 0 as $n\to \infty$. One of the main tools for proving these conditional limit laws is the \textit{uniform} quenched functional Central Limit Theorem, that states that the convergence is uniform with respect to the starting point, provided that the starting point is chosen in a certain interval around the origin.

This paper was updated and split in two parts: see http://arxiv.org/abs/1210.0951 and http://arxiv.org/abs/1210.0591

References in corpus (2)

• Invariance principles for random walks conditioned to stay positive
• Quenched invariance principle for the Knudsen stochastic billiard in a random tube

Cited by in corpus (1)

• Recent progress on the Random Conductance Model