Electronic Journal of Probability

Cut Times for Simple Random Walk

Gregory Lawler

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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

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

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

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60J15

Random walk cut points intersection exponent

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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

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  • 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.