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

Lawler, Gregory F.

Electron. J. Probab. 1, No. 13, Paper 13, 24 p. (1996).

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