Involve: A Journal of Mathematics

  • Involve
  • Volume 12, Number 1 (2019), 97-115.

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

Full-text: Access denied (no subscription detected)

However, an active subscription may be available with MSP at msp.org/involve.

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text

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 = lim n ( X n n ) , 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.

Article information

Source
Involve, Volume 12, Number 1 (2019), 97-115.

Dates
Received: 10 July 2017
Revised: 9 November 2017
Accepted: 10 December 2017
First available in Project Euclid: 26 October 2018

Permanent link to this document
https://projecteuclid.org/euclid.involve/1540519237

Digital Object Identifier
doi:10.2140/involve.2019.12.97

Mathematical Reviews number (MathSciNet)
MR3810481

Zentralblatt MATH identifier
1391.60239

Subjects
Primary: 60K35: Interacting random processes; statistical mechanics type models; percolation theory [See also 82B43, 82C43]
Secondary: 60G50: Sums of independent random variables; random walks

Keywords
excited random walk Markov chain stationary distribution

Citation

Madden, Erin; Kidd, Brian; Levin, Owen; Peterson, Jonathon; Smith, Jacob; Stangl, Kevin M. Upper and lower bounds on the speed of a one-dimensional excited random walk. Involve 12 (2019), no. 1, 97--115. doi:10.2140/involve.2019.12.97. https://projecteuclid.org/euclid.involve/1540519237


Export citation

References

  • A.-L. Basdevant and A. Singh, “On the speed of a cookie random walk”, Probab. Theory Related Fields 141:3-4 (2008), 625–645.
  • A.-L. Basdevant and A. Singh, “Rate of growth of a transient cookie random walk”, Electron. J. Probab. 13 (2008), 811–851.
  • I. Benjamini and D. B. Wilson, “Excited random walk”, Electron. Comm. Probab. 8 (2003), 86–92.
  • E. Kosygina and M. P. W. Zerner, “Positively and negatively excited random walks on integers, with branching processes”, Electron. J. Probab. 13 (2008), 1952–1979.
  • G. Pólya, “Über eine Aufgabe der Wahrscheinlichkeitsrechnung betreffend die Irrfahrt im Straß ennetz”, Math. Ann. 84:1-2 (1921), 149–160.
  • M. P. W. Zerner, “Multi-excited random walks on integers”, Probab. Theory Related Fields 133:1 (2005), 98–122.