  We study the excited random walk, in which a walk that is at a site that
contains cookies eats one cookie and then hops to the right with probability p
and to the left with probability q=1-p. If the walk hops onto an empty site,
there is no bias. For the 1-excited walk on the half-line (one cookie initially
at each site), the probability of first returning to the starting point at time
t scales as t^{-(2-p)}. Although the average return time to the origin is
infinite for all p, the walk eats, on average, only a finite number of cookies
until this first return when p<1/2. For the infinite line, the probability
distribution for the 1-excited walk has an unusual anomaly at the origin. The
positions of the leftmost and rightmost uneaten cookies can be accurately
estimated by probabilistic arguments and their corresponding distributions have
power-law singularities near 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>1/2
dependent on p, and a breakdown of scaling for the probability distribution of
the walk.
