Excited Random Walk in One Dimension

We study the $k$-excited random walk, in which each site initially contains $k$ cookies, and a random walk that is at a site that contains at least one cookie eats a cookie and then hops to the right with probability $p$ and to the left with probability $q=1-p$. If the walk hops from an empty site, there is no bias. For the 1-excited walk on the half-line (each site initially contains one cookie), the probability of first returning to the starting point at time $t$ scales as $t^{-1-q}$. We also derive the probability distribution of the position of the leftmost uneaten cookie in the large time limit. For the infinite line, the probability distribution of the position of the 1-excited walk has an unusual anomaly at the origin and the distributions of positions for the leftmost and rightmost uneaten cookie develop a power-law singularity at the origin. The 2-excited walk on the infinite line exhibits peculiar features in the regime $p>3/4$, where the walk is transient, including a mean displacement that grows as $t^\nu$, with $\nu>\frac{1}{2}$ dependent on $p$, and a breakdown of scaling for the probability distribution of the walk.