The Annals of Probability

Recurrence of random walk traces

Itai Benjamini, Ori Gurel-Gurevich, and Russell Lyons

Full-text: Open access


We show that the edges crossed by a random walk in a network form a recurrent graph a.s. In fact, the same is true when those edges are weighted by the number of crossings.

Article information

Ann. Probab., Volume 35, Number 2 (2007), 732-738.

First available in Project Euclid: 30 March 2007

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60J05: Discrete-time Markov processes on general state spaces
Secondary: 60D05: Geometric probability and stochastic geometry [See also 52A22, 53C65]

Paths networks graphs


Benjamini, Itai; Gurel-Gurevich, Ori; Lyons, Russell. Recurrence of random walk traces. Ann. Probab. 35 (2007), no. 2, 732--738. doi:10.1214/009117906000000935.

Export citation


  • Ancona, A., Lyons, R. and Peres, Y. (1999). Crossing estimates and convergence of Dirichlet functions along random walk and diffusion paths. Ann. Probab. 27 970--989.
  • Benjamini, I. and Gurel-Gurevich, O. (2005). Almost sure recurrence of the simple random walk path. Unpublished manuscript. Available at arXiv:math.PR/0508270.
  • Benjamini, I. and Peres, Y. (1994). Markov chains indexed by trees. Ann. Probab. 22 219--243.
  • Benjamini, I. and Peres, Y. (1994). Tree-indexed random walks on groups and first passage percolation. Probab. Theory Related Fields 98 91--112.
  • Benjamini, I. and Schramm, O. (2001). Recurrence of distributional limits of finite planar graphs. Electron. J. Probab. 6 1--13.
  • Grigor'yan, A. (1999). Analytic and geometric background of recurrence and non-explosion of the Brownian motion on Riemannian manifolds. Bull. Amer. Math. Soc. (N.S.) 36 135--249.
  • James, N. and Peres, Y. (1996). Cutpoints and exchangeable events for random walks. Teor. Veroyatnost. i Primenen. 41 854--868.
  • Lawler, G. F. (1996). Cut times for simple random walk. Electron. J. Probab. 1 1--24.
  • Lyons, R. and Peres, Y. (2007). Random walks with finitely many cut-times. Unpublished manuscript.
  • Lyons, R. with Peres, Y. (2007). Probability on Trees and Networks. Cambridge Univ. Press. To appear. Current version available at
  • Morris, B. (2003). The components of the wired spanning forest are recurrent. Probab. Theory Related Fields 125 259--265.
  • Nash-Williams, C. \relax St. J. A. (1959). Random walk and electric currents in networks. Proc. Cambridge Philos. Soc. 55 181--194.