From Coalescing Random Walks on a Torus to Kingman's Coalescent

Let T-N(d), d >= 2, be the discrete d-dimensional torus with N-d points. Place a particle at each site of T-N(d) 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 theta(N) for which C-N/theta(N) converges to the sum of independent exponential random variables. Denote by Z(t)(N) the total number of particles at time t. We prove that the sequence of Markov chains (Z(t theta N)(N))(t >= 0) converges to the total number of partitions in Kingman's coalescent.