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1996 Cut Times for Simple Random Walk
Gregory Lawler
Author Affiliations +
Electron. J. Probab. 1: 1-24 (1996). DOI: 10.1214/EJP.v1-13


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.


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Gregory Lawler. "Cut Times for Simple Random Walk." Electron. J. Probab. 1 1 - 24, 1996.


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

zbMATH: 0888.60059
MathSciNet: MR1423466
Digital Object Identifier: 10.1214/EJP.v1-13

Primary: 60J15

Keywords: cut points , intersection exponent , Random walk

Vol.1 • 1996
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