Excited Mob

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.