The excited random walk in one dimension

Abstract

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 ν , with ν > 1 2 dependent on p, and a breakdown of scaling for the probability distribution of the walk.

Monotonicity for excited random walk in high dimensions

“…Proof Proceeding exactly as in the proof of Proposition 3.2, except that the bound on | k | is missing the β term, we obtain χ (1) …”

Trading cookies in a gambler’s ruin scenario

“…It is natural to study this type of problem within a game-theoretic framework, where exact features of the reinforcing mechanism are determined through the interaction between the walker and a supplier. This is in contrast to the usual excited or cookie random walk [Antal and Redner 2005;Zerner 2005] (see [Menshikov et al 2012] for an up-to-date review and references), where the walker, as a pricetaker in a large market, has no effect on determining the parameters of the cookie environment.…”

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“…It is natural to study this type of problem within a game-theoretic framework, where exact features of the reinforcing mechanism are determined through the interaction between the walker and a supplier. This is in contrast to the usual excited or cookie random walk [Antal and Redner 2005;Zerner 2005] (see [Menshikov et al 2012] for an up-to-date review and references), where the walker, as a pricetaker in a large market, has no effect on determining the parameters of the cookie environment.…”

“…These quantities are fundamental for the random walk theory. They have been discussed in [Antal and Redner 2005], based on arguments of a different type from ours.…”

“…In what follows we focus on finding the equilibrium price for a cookie when X 0 = 1 and exactly one cookie is placed at each integer site within the interval (0, b). The corresponding random walk (X k ) k≥0 is usually referred to as the 1-excited random walk on ‫ޚ‬ (see, for instance, [Antal and Redner 2005;Benjamini and Wilson 2003]). Our main results in this section are stated in Theorems 7.1 and 7.2; see also two remarks concluding the section.…”