## Cut times for simple random walk.(English)Zbl 0888.60059

Summary: Let $$S(n)$$ be a simple random walk taking values in $$\mathbb{Z}^d$$. 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 $P\{S[0,n] \cap S[n+1,2n] = \emptyset \} \asymp n^{-\zeta},$ where $$\asymp$$ denotes that each side is bounded by a constant times the other side.

### MSC:

 60G50 Sums of independent random variables; random walks

### Keywords:

random walk; cut points; intersection exponent
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