Electronic Communications in Probability

Recurrence and transience of excited random walks on $Z^d$ and strips

Martin Zerner

Full-text: Open access


We investigate excited random walks on $Z^d, d\ge 1,$ and on planar strips $Z\times{0,1,\ldots,L-1}$ which have a drift in a given direction. The strength of the drift may depend on a random i.i.d. environment and on the local time of the walk. We give exact criteria for recurrence and transience, thus generalizing results by Benjamini and Wilson for once-excited random walk on $Z^d$ and by the author for multi-excited random walk on $Z$.

Article information

Electron. Commun. Probab., Volume 11 (2006), paper no. 12, 118-128.

Accepted: 7 July 2006
First available in Project Euclid: 4 June 2016

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60K35: Interacting random processes; statistical mechanics type models; percolation theory [See also 82B43, 82C43]
Secondary: 60K37: Processes in random environments 60J10: Markov chains (discrete-time Markov processes on discrete state spaces)

Excited Random Walk Recurrence Self-Interacting Random Walk Transience

This work is licensed under aCreative Commons Attribution 3.0 License.


Zerner, Martin. Recurrence and transience of excited random walks on $Z^d$ and strips. Electron. Commun. Probab. 11 (2006), paper no. 12, 118--128. doi:10.1214/ECP.v11-1200. https://projecteuclid.org/euclid.ecp/1465058855

Export citation


  • G. Amir, I. Benjamini and G. Kozma. Excited random walk against a wall. Preprint available at http://arxiv.org/abs/math.PR/0509464 (2005).
  • T. Antal and S. Redner. The excited random walk in one dimension. J. Phys. A: Math. Gen. 38 (2005), 2555–2577.
  • I. Benjamini and D.B. Wilson. Excited random walk. Elect. Comm. Probab. 8 (2003), 86–92.
  • G. Kozma. Excited random walk in three dimensions has positive speed. Preprint available at http://arxiv.org/abs/math.PR/0310305 (2003).
  • G. Kozma. Excited random walk in two dimensions has linear speed. Preprint available at http://arxiv.org/abs/math.PR/0512535 (2005).
  • S. P. Meyn and R. L. Tweedie. Markov Chains and Stochastic Stability. (1993) Springer.
  • A.G. Pakes. On the critical Galton-Watson process with immigration. J. Austral. Math. Soc. 12 (1971), 476–482.
  • T. Sellke. Reinforced random walk on the d-dimensional integer lattice. Technical report #94-26, Dept. of Statistics, Purdue University. (1994)
  • A.-S. Sznitman and M.P.W. Zerner. A law of large numbers for random walks in random environment. Ann. Probab. 27, No. 4 (1999), 1851–1869.
  • M.P.W. Zerner. Multi-excited random walks on integers. Probab. Theory Related Fields 133 (2005), 98–122.
  • M.P.W. Zerner and F. Merkl. A zero-one law for planar random walks in random environment. Ann. Probab. 29 (2001), 1716–1732.
  • A. Zubkov. The life spans of a branching process with immigration. Theory Prob. Appl. 17 (1972), 174–183.