## Electronic Journal of Probability

### Cut Times for Simple Random Walk

Gregory Lawler

#### 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

Rights

#### 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

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