A monotonicity property for random walk in a partially random environment

TL;DR

This paper establishes a law of large numbers for certain i.i.d. random walks in multi-dimensional environments and demonstrates a monotonicity property of the walk's speed using advanced probabilistic techniques.

Contribution

It extends previous results by proving a law of large numbers and a new monotonicity property for the speed of random walks in specific random environments.

Findings

Law of large numbers for random walks in i.i.d. environments

Monotonicity of the first coordinate of the walk's speed

Application of lace expansion to self-interacting walks

Abstract

We prove a law of large numbers for random walks in certain kinds of i.i.d. random environments in Z^d that is an extension of a result of Bolthausen, Sznitman and Zeitouni (2003). We use this result, along with the lace expansion for self-interacting random walks, to prove a monotonicity result for the first coordinate of the speed of the random walk under some strong assumptions on the distribution of the environment.