Limiting Point Processes for Rescalings of Coalescing and Annihilating Random Walks on Zᵈ

Let $p(x, y)$ be an arbitrary random walk on Zᵈ. Let ξₜ be the system of coalescing random walks based on p, starting with all sites occupied, and let ηₜ be the corresponding system of annihilating random walks. The spatial rescalings P(0 ∈ ξₜ)1/dξₜ for t 0 form a tight family of point processes on Rᵈ. Any limiting point process as t →∞ has Lesbegue measure as its intensity, and has no multiple points. When p is simple random walk on Zᵈ these rescalings converge in distribution, to the simple Poisson point process for d ≥ 2, and to a non-Poisson limit for $d = 1$. For a large class of p, we prove that P(0 ∈ ηₜ)/P(0 ∈ ξₜ) → 1/2 as t →∞. A generalization of this result, proved for nearest neighbor random walks on Z¹, and for all multidimensional p, implies that the limiting point process for rescalings P(0 ∈ ξₜ)1/dηₜ of the system of annihilating random walks is the one half thinning of the limiting point process for the corresponding coalescing system.