Cut Times for Simple Random Walk

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.