From coalescing random walks on a torus to Kingman’s coalescent

Let TNd, d≥ 2 , be the discrete d-dimensional torus with N^d points. Place a particle at each site of TNd and let them evolve as independent, nearest-neighbor, symmetric, continuous-time random walks. Each time two particles meet, they coalesce into one. Denote by C_N the first time the set of particles is reduced to a singleton. Cox (Ann Probab 17:1333–1366, 1989) proved the existence of a time-scale θ_N for which C_N/ θ_N converges to the sum of independent exponential random variables. Denote by ZtN the total number of particles at time t. We prove that the sequence of Markov chains (ZtθNN)t≥0 converges to the total number of partitions in Kingman’s coalescent.