Electronic Journal of Probability

Cut Times for Simple Random Walk

Gregory Lawler

Full-text: Open access

Abstract

Let $S(n)$ be a simple random walk taking values in $Z^d$. A time $n$ is called a cut time if \[ S[0,n] \cap S[n+1,\infty) = \emptyset . \] We show that in three dimensions the number of cut times less than $n$ grows like $n^{1 - \zeta}$ where $\zeta = \zeta_d$ is the intersection exponent. As part of the proof we show that in two or three dimensions \[ P(S[0,n] \cap S[n+1,2n] = \emptyset ) \sim n^{-\zeta}, \] where $\sim$ denotes that each side is bounded by a constant times the other side.

Article information

Source
Electron. J. Probab., Volume 1 (1996), paper no. 13, 24 pp.

Dates
Accepted: 19 October 1996
First available in Project Euclid: 25 January 2016

Permanent link to this document
https://projecteuclid.org/euclid.ejp/1453756476

Digital Object Identifier
doi:10.1214/EJP.v1-13

Mathematical Reviews number (MathSciNet)
MR1423466

Zentralblatt MATH identifier
0888.60059

Subjects
Primary: 60J15

Keywords
Random walk cut points intersection exponent

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

Citation

Lawler, Gregory. Cut Times for Simple Random Walk. Electron. J. Probab. 1 (1996), paper no. 13, 24 pp. doi:10.1214/EJP.v1-13. https://projecteuclid.org/euclid.ejp/1453756476


Export citation

References

  • Ahlfors, L. (1973). Conformal Invariants. Topics in Geometric Function Theory. McGraw Hill
  • Breiman, L. (1992) Probability. SIAM
  • Burdzy, K. and Lawler, G. (1990). Non-intersection exponents for random walk and Brownian motion. Part I: Existence and an invariance principle. Probab. Th. and Rel. Fields 84 393-410.
  • Burdzy, K. and Lawler, G. (1990). Non-intersection exponents for random walk and Brownian motion. Part II: Estimates and applications to a random fractal. Ann. Probab. 18 981–1009.
  • Burdzy, K., Lawler, G., and Polaski, T. (1989). On the critical exponent for random walk intersections. J. Stat. Phys. 56 1–12.
  • Cranston, M. and Mountford, T. (1991). An extension of a result of Burdzy and Lawler. Probab. Th. and Rel. Fields 89, 487-502.
  • Duplantier, B. and Kwon, K.-H. (1988). Conformal invariance and intersections of random walks. Phys. Rev. Lett. 61 2514-2517.
  • James, N. and Peres, Y. (1995). Cutpoints and exchangeable events for random walks, to appear in Theory of Probab. and Appl.
  • Lawler, G. (1991). Intersections of Random Walks. Birkhauser-Boston
  • Lawler, G. (1992). Escape probabilities for slowly recurrent sets. Probab. Th. and Rel. Fields 94, 91-117.
  • Lawler, G. (1996). Hausdorff dimension of cut points for Brownian motion. Electon. J. Probab. 1, paper no. 2.
  • Lawler, G. and Puckette, E (1994), The disconnection exponent for simple random walk, to appear in Israel Journal of Mathematics.
  • Li, B. and Sokal, A. (1990). High-precision Monte Carlo test of the conformal-invariance predictions for two-dimensional mutually avoiding walks. J. Stat. Phys. 61 723-748.