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

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 = limn(Xnn), where Xn 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.

excited random walk, Markov chain, stationary distribution
Mathematical Subject Classification 2010
Primary: 60K35
Secondary: 60G50
Received: 10 July 2017
Revised: 9 November 2017
Accepted: 10 December 2017
Published: 31 May 2018

Communicated by John C. Wierman
Erin Madden
Eastern Illinois University
Charleston, IL
United States
Department of Mathematics
University of Illinois at Urbana-Champaign
Champaign, IL
United States
Brian Kidd
Purdue University
West Lafayette, IN
United States
Department of Statistics
Texas A&M University
College Station, TX
United States
Owen Levin
University of Minnesota
Minneapolis, MN
United States
Jonathon Peterson
Department of Mathematics
Purdue University
West Lafayette, IN
United States
Jacob Smith
Franklin College
Franklin, IN
United States
Department of Mathematical Sciences
University of Cincinnati
Cincinnati, OH
United States
Kevin M. Stangl
University of California
Los Angeles, CA
United States
Toyota Technological Institute at Chicago
Chicago, IL
United States