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

TL;DR

This paper demonstrates that the coalescing random walks on a high-dimensional torus, when properly scaled, converge to Kingman's coalescent, linking spatial random processes to classical coalescent theory.

Contribution

It establishes the convergence of coalescing random walks on a torus to Kingman's coalescent under appropriate time scaling, extending the understanding of spatial coalescent processes.

Findings

Convergence of scaled coalescing random walks to Kingman's coalescent.

Identification of the time-scale $ heta_N$ for convergence.

Demonstration of the limiting behavior of the number of particles.

Abstract

Let $\mathbb{T}^d_N$, $d\ge 2$, be the discrete $d$-dimensional torus with $N^d$ points. Place a particle at each site of $\mathbb{T}^d_N$ 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 $Z^N_t$ the total number of particles at time $t$. We prove that the sequence of Markov chains $(Z^N_{t\theta_N})_{t\ge 0}$ converges to the total number of partitions in Kingman's coalescent.