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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
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Abstract |
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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 , where is the state of the walk at time . 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. |
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Keywords
excited random walk, Markov chain, stationary distribution
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Mathematical Subject Classification 2010
Primary: 60K35
Secondary: 60G50
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Milestones
Received: 10 July 2017
Revised: 9 November 2017
Accepted: 10 December 2017
Published: 31 May 2018
Communicated by John C. Wierman
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Authors |
| 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 |
