Poissonization-based collision threshold derivation for random walks on lattices

TL;DR

The paper presents a self-contained, Poissonization-based derivation of the collision threshold for two independent simple random walks on $\mathbb{Z}^d$. By analyzing a continuous-time difference walk and expressing the collision probability through the modified Bessel function $I_0$, it derives a precise asymptotic decay $P(D(t)=0) \sim C t^{-d/2}$, with $C=(\frac{d}{\pi})^{d/2}$. This leads to a convergent expected number of collisions when $d \ge 3$ and a divergent one when $d \le 2$, thereby re-establishing the dimension threshold for collision finiteness and computing a leading constant for the decay. The approach bridges discrete and continuous-time perspectives and provides a general framework that could extend to related processes such as Brownian motion limits and voter-model dynamics.

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.

Theorems & Definitions (17)

• Theorem 1.1
• Theorem 1.2
• Definition 2.1
• Definition 2.2
• Lemma 2.3
• proof
• Proposition 2.4
• proof
• Definition 3.1
• Proposition 3.2