Crossing velocities for an annealed random walk in a random potential

TL;DR

This paper studies a random walk in a random potential, demonstrating linear growth of expected hitting times and establishing positive asymptotic speed in one dimension, advancing understanding of such stochastic processes.

Contribution

It proves linear growth of expected hitting times and confirms positive asymptotic speed in one dimension for annealed random walks in random potentials.

Findings

Expected hitting time grows linearly with distance

Existence of positive asymptotic speed in one dimension

Results apply to i.i.d. non-negative potentials

Abstract

We consider a random walk in an i.i.d. non-negative potential on the d-dimensional integer lattice. The walk starts at the origin and is conditioned to hit a remote location y on the lattice. We prove that the expected time under the annealed path measure needed by the random walk to reach y grows only linearly in the distance from y to the origin. In dimension one we show the existence of the asymptotic positive speed.