Poissonization-based collision threshold derivation for random walks on lattices

TL;DR

This paper presents a Poissonization-based method to derive the collision threshold for two independent random walks on integer lattices, showing finiteness of expected collisions depends on the dimension.

Contribution

It introduces a novel Poissonization technique to analyze collision probabilities, providing a clear proof and a general asymptotic formula for collisions in lattice random walks.

Findings

Expected number of collisions is finite if and only if dimension d ≥ 3.

Derived a general asymptotic formula for the number of collisions.

Provided a self-contained proof using Bessel functions and asymptotic analysis.

Abstract

In this expository note, we give a short derivation of the expected number of collisions between two independent simple random walkers on integer lattices. Adapting a Poissonization technique introduced by Lange, we express the collision probability as the return probability of the continuous-time difference walk, given by a modified Bessel function. Analyzing its asymptotic decay yields a clean, self-contained proof that the expected number of collisions in $\mathbb{Z}^d$ is finite if and only if $d\geq3$. We also provide a general formula for the asymptotic number of collisions.