Electronic Journal of Probability

Two-sided random walks conditioned to have no intersections

Daisuke Shiraishi

Full-text: Open access


Let $S^{1},S^{2}$ be independent simple random walks in $\mathbb{Z}^{d}$ ($d=2,3$) started at the origin. We construct two-sided random walk paths conditioned that $S^{1}[0,\infty ) \cap S^{2}[1, \infty ) = \emptyset$ by showing the existence of the following limit:\begin{equation*}\lim _{n \rightarrow \infty } P ( \cdot   |  S^{1}[0, \tau ^{1} ( n) ] \cap S^{2}[1, \tau ^{2}(n) ] = \emptyset ),\end{equation*}where $\tau^{i}(n) = \inf \{ k \ge 0 : |S^{i} (k) | \ge n \}$. Moreover, we give upper bounds of the rate of the convergence. These are discrete analogues of results for Brownian motion obtained by Lawler.

Article information

Electron. J. Probab., Volume 17 (2012), paper no. 18, 24 pp.

Accepted: 28 February 2012
First available in Project Euclid: 4 June 2016

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 05C81: Random walks on graphs

Random walks Cut points Invariant measure

This work is licensed under aCreative Commons Attribution 3.0 License.


Shiraishi, Daisuke. Two-sided random walks conditioned to have no intersections. Electron. J. Probab. 17 (2012), paper no. 18, 24 pp. doi:10.1214/EJP.v17-1742. https://projecteuclid.org/euclid.ejp/1465062340

Export citation


  • Burdzy, Krzysztof; Lawler, Gregory F. Nonintersection exponents for Brownian paths. II. Estimates and applications to a random fractal. Ann. Probab. 18 (1990), no. 3, 981–1009.
  • Lawler, Gregory F. Intersections of random walks. Probability and its Applications. Birkhäuser Boston, Inc., Boston, MA, 1991. 219 pp. ISBN: 0-8176-3557-2.
  • Lawler, Gregory F. Nonintersecting planar Brownian motions. Math. Phys. Electron. J. 1 (1995), Paper 4, approx. 35 pp. (electronic).
  • Lawler, Gregory F. Cut times for simple random walk. Electron. J. Probab. 1 (1996), no. 13, approx. 24 pp. (electronic).
  • Lawler, Gregory F. Hausdorff dimension of cut points for Brownian motion. Electron. J. Probab. 1 (1996), no. 2, approx. 20 pp. (electronic).
  • Lawler, Gregory F.; Schramm, Oded; Werner, Wendelin. Values of Brownian intersection exponents. II. Plane exponents. Acta Math. 187 (2001), no. 2, 275–308.
  • Lawler, Gregory F. Conformally invariant processes in the plane. Mathematical Surveys and Monographs, 114. American Mathematical Society, Providence, RI, 2005. xii+242 pp. ISBN: 0-8218-3677-3.
  • Gregory F. Lawler, Brigitta Vermesi. Fast convergence to an invariant measure for non-intersecting 3-dimensional Brownian paths. (2010) preprint, available at http://arxiv.org/abs/1008.4830
  • Daisuke Shiraishi. Subdiffusive behavior of random walk on conditioned two-sided random walks in two dimensions. preprint