Electronic Communications in Probability

Excited Random Walk

Itai Benjamini and David Wilson

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A random walk on $\mathbb{Z}^d$ is excited if the first time it visits a vertex there is a bias in one direction, but on subsequent visits to that vertex the walker picks a neighbor uniformly at random. We show that excited random walk on $\mathbb{Z}^d$ is transient iff $d \gt 1$.

Article information

Electron. Commun. Probab., Volume 8 (2003), paper no. 9, 86-92.

Accepted: 24 June 2003
First available in Project Euclid: 18 May 2016

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60J10: Markov chains (discrete-time Markov processes on discrete state spaces)

Perturbed random walk transience

This work is licensed under aCreative Commons Attribution 3.0 License.


Benjamini, Itai; Wilson, David. Excited Random Walk. Electron. Commun. Probab. 8 (2003), paper no. 9, 86--92. doi:10.1214/ECP.v8-1072. https://projecteuclid.org/euclid.ecp/1463608893

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  • O. Angel, I. Benjamini, and B. Virag. Random walks that avoid their past convex hull. Elec. Comm. Prob. 8(2):6-16, 2003.
  • R. F. Bass and T. Kumagai. Laws of the iterated logarithm for the range of random walks in two and three dimensions. Ann. Prob. 30:1369–1396, 2002.
  • M. Bousquet-Mélou and G. Schaeffer. Walks on the slit plane. Probab. Theory Related Fields 124(3):305–344, 2002. arXiv:math.CO/0012230.
  • B. Davis. Reinforced random walk. Probab. Theory Related Fields 84(2):203–229, 1990.
  • B. Davis. Weak limits of perturbed Brownian motion and the equation $Y_t = B_t + \alpha\sup\{Y_s\colon s \leq t\}+\beta\inf\{Y\sb s\colon s \leq t\}$. Ann. Prob. 24:2007–2023, 1996.
  • B. Davis. Brownian motion and random walk perturbed at extrema. Probab. Theory Related Fields 113(4):501–518, 1999.
  • P. G. Doyle and J. L. Snell. Random Walks and Electric Networks, Mathematical Association of America, 1984. arXiv:math.PR/0001057.
  • R. Durrett. Probability: Theory and Examples, second edition. Duxbury Press, 1996. 503 pp.
  • A. Dvoretzky and P. Erdős. Some problems on random walk in space. Proc. 2nd Berkeley Symp., pp. 353–367, 1951.
  • M. L. Glasser and I. J. Zucker. Extended Watson integrals for the cubic lattice. Proc. Natl. Acad. Sci., USA 74:1800-1801, 1977.
  • G. F. Lawler. Intersections of Random Walks. Probability and its Applications. Birkhäuser, Boston, MA, 1991. 219 pp.
  • G. F. Lawler. A lower bound on the growth exponent for loop-erased random walk in two dimensions. ESAIM Probab. Statist. 3:1–21, 1999.
  • R. Pemantle. Random processes with reinforcement. Preprint, 28 pp. URL: http://www.math.ohio-state.edu/~pemantle/papers/Papers.html.
  • M. Perman and W. Werner. Perturbed Brownian motions. Probab. Theory Related Fields 108:357–383, 1997.
  • O. Schramm. Scaling limits of loop-erased random walks and uniform spanning trees. Israel J. Math. 118:221–288, 2000. arXiv:math.PR/9904022.
  • B. Tóth and W. Werner. The true self-repelling motion. Probab. Theory Related Fields 111(3):375–452, 1998.