Limiting velocity of high-dimensional random walk in random environment

TL;DR

This paper proves that in high-dimensional uniformly elliptic i.i.d. environments, a random walk has at most one non-zero limiting velocity, confirming a law of large numbers and linking various conjectures.

Contribution

It establishes the uniqueness of the non-zero limiting velocity for high-dimensional random walks in random environments, advancing understanding of their asymptotic behavior.

Findings

At most one non-zero limiting velocity in dimensions ≥ 5

Law of large numbers proven in symmetric cases

Connections made between different conjectures in the field

Abstract

We show that random walk in uniformly elliptic i.i.d. environment in dimension $\geq5$ has at most one non zero limiting velocity. In particular this proves a law of large numbers in the distributionally symmetric case and establishes connections between different conjectures.