Mean field asymptotics for the planar stepping stone model

Let ξ, denote coalescing random walks, ς, the stepping stone model, on the two-dimensional integer lattice ℤ². We prove four theorems, announced previously in [4] and [5], concerning these evolutions. Theorem 1 states that if we spread out the initial locations of m coalescing walks, then, after appropriate rescaling, the partition of (1, 2,., m) that records which sets of particles have coalesced by time s converges to Kingman's coalescent π(s) [10]. Various duality equations then lead to results for the stepping stone model. Theorem 2 identifies an exchangeable limit distribution for power-law thinnings of ς,. Theorem 3, our main result, says that appropriately scaled block density processes derived from ς, converge to a time change of the Wright-Fisher diffusion. We treat stepping stone models with finitely many possible types per site, and also the case in which every site of ℤ²has a different type initially. Finally, Theorem 4 describes the asymptotic number of types represented in a large box centred at the origin, generalizing some results from [1].