Electronic Journal of Probability

Two-sided random walks conditioned to have no intersections

Daisuke Shiraishi

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Abstract

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

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

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

Permanent link to this document
https://projecteuclid.org/euclid.ejp/1465062340

Digital Object Identifier
doi:10.1214/EJP.v17-1742

Mathematical Reviews number (MathSciNet)
MR2900459

Zentralblatt MATH identifier
1244.05209

Subjects
Primary: 05C81: Random walks on graphs

Keywords
Random walks Cut points Invariant measure

Rights
This work is licensed under aCreative Commons Attribution 3.0 License.

Citation

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


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References

  • 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