Poissonization-based collision threshold derivation for random walks on lattices

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.