Excited Mob

TL;DR

This paper investigates the behavior of excited random walks and mobs in stationary ergodic cookie environments, establishing conditions for transience and positive speed, and providing the first example of such environments with positive speed without trivial comparison.

Contribution

It introduces new criteria for transience and positive speed of excited random walks in stationary ergodic environments, including the first example with positive speed not derived from trivial i.i.d. comparison.

Findings

Right transience if and only if δ > k+1

Positive speed if and only if δ > k+2

Law of large numbers and zero-one law for directional transience

Abstract

We show that for an i.i.d. bounded and weakly elliptic cookie environment, one dimensional excited random walk on the $k$-time leftover environment is right transient if and only if $\delta > k+1$ and has positive speed if and only if $\delta > k+2$, where $\delta$ is the expected drift per site. This gives, to the best of our knowledge, the first example of an environment with positive speed that has stationary and ergodic properties but does not follow by trivial comparison to an i.i.d. environment. In another formulation, we show that on such environments an excited mob of $k$ walkers is right transient if and only if $\delta > k$ and moves with positive speed if and only if $\delta>k+1$. We show that for stationary and erogodic cookie environments, a law of large numbers for the walkers on leftover environments holds. Whenever the environments are also elliptic, a zero-one law for directional transience is also proven.