Ballisticity of Random walks in Random Environments on $\mathbb{Z}$ with Bounded Jumps

TL;DR

This paper characterizes the conditions for ballistic behavior of i.i.d. random walks with bounded jumps in random environments on , providing formulas for limiting speed without requiring uniform ellipticity.

Contribution

It introduces new characterizations of ballisticity that do not rely on uniform ellipticity, extending understanding to bounded jump scenarios.

Findings

Provided formulas for limiting speed in the non-nearest-neighbor case.

Characterized ballisticity without uniform ellipticity conditions.

Connected results to previous work on Dirichlet environments.

Abstract

We characterize ballistic behavior for general i.i.d. random walks in random environments on $\mathbb{Z}$ with bounded jumps. The two characterizations we provide do not use uniform ellipticity conditions. They are natural in the sense that they both relate to formulas for the limiting speed in the nearest-neighbor case. Note: This paper duplicates results from some versions of the preprint "Random walks in Dirichlet random environments on $\mathbb{Z}$ with bounded jumps." (arxiv: 2104.14950). The present paper is being split off for reasons of length, and the plan is to remove these results from a future version of the previous paper and replace them with a citation of the present preprint.