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.
展开▼
机译：令$ S（n）$是一个简单的随机游走，取值$ Z ^ d $。如果 [S [0，n] cap S [n + 1， infty）= emptyset，则时间$ n $被称为切割时间。 ]我们显示，在三个维度中，小于$ n $的剪切时间的数量增长为$ n ^ {1- zeta} $，其中$ zeta = zeta_d $是交点指数。作为证明的一部分，我们显示在二维或三维中 [P（S [0，n] cap S [n + 1,2n] = emptyset） sim n ^ {- zeta}，]其中$ sim $表示每一边都被常数乘以另一边。
展开▼
