Vol. 12, No. 1, 2019

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Upper and lower bounds on the speed of a one-dimensional excited random walk

Erin Madden, Brian Kidd, Owen Levin, Jonathon Peterson, Jacob Smith and Kevin M. Stangl

Vol. 12 (2019), No. 1, 97–115
Abstract

An excited random walk (ERW) is a self-interacting non-Markovian random walk in which the future behavior of the walk is influenced by the number of times the walk has previously visited its current site. We study the speed of the walk, defined as $V=\underset{n\to \infty }{lim}\left({X}_{n}∕n\right)$, where ${X}_{n}$ is the state of the walk at time $n$. While results exist that indicate when the speed is nonzero, there exists no explicit formula for the speed. It is difficult to solve for the speed directly due to complex dependencies in the walk since the next step of the walker depends on how many times the walker has reached the current site. We derive the first nontrivial upper and lower bounds for the speed of the walk. In certain cases these upper and lower bounds are remarkably close together.

Keywords
excited random walk, Markov chain, stationary distribution
Primary: 60K35
Secondary: 60G50