Excited against the tide: A random walk with competing drifts

Abstract

We study a random walk that has a drift $\frac{\beta}{d}$ to the right when located at a previously unvisited vertex and a drift $\frac{\mu}{d}$ to the left otherwise. We prove that in high dimensions, for every $\mu$, the drift to the right is a strictly increasing and continuous function of $\beta$, and that there is precisely one value $\beta_0(\mu,d)$ for which the resulting speed is zero.