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

Abstract

Let ${\mathbb{T}}_N^d$, $d\ge 2$, be the discrete $d$-dimensional torus with $N^d$ points. Place a particle at each site of ${\mathbb{T}}_N^d$ 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 [6] 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 $(Z_{t{\theta }_N}^N{)}_{t\ge 0}$ converges to the total number of partitions in Kingman's coalescent.