Acta Mathematica Hungarica

, Volume 96, Issue 3, pp 187–220

# Self-intersections of random walks on lattices

• Xian Yin Zhou
Article

## Abstract

Let {Xnd}n≥0be a uniform symmetric random walk on Zd, and Π(d) (a,b)={Xnd ∈ Zd : a ≤ n ≤ b}. Suppose f(n) is an integer-valued function on n and increases to infinity as n↑∞, and let
$$E_n^{\left( d \right)} = \left\{ {\prod {^{\left( d \right)} } \left( {0,n} \right) \cap \prod {^{\left( d \right)} } \left( {n + f\left( n \right),\infty } \right) \ne \emptyset } \right\}$$
Estimates on the probability of the event $$E_n^{\left( d \right)}$$ are obtained for $$d \geqq 3$$. As an application, a necessary and sufficient condition to ensure $$P\left( {E_n^{\left( d \right)} ,{\text{i}}{\text{.o}}{\text{.}}} \right) = 0\quad {\text{or}}\quad {\text{1}}$$ is derived for $$d \geqq 3$$. These extend some results obtained by Erdős and Taylor about the self-intersections of the simple random walk on Zd.
random walk self-intersection hitting time Green function range

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