Recovering the Brownian Coalescent Point Process from the Kingman Coalescent by Conditional Sampling

TL;DR

This paper demonstrates how the genealogy of a sample from a population conditioned on recent common ancestry converges to a Brownian coalescent point process, linking Kingman coalescent and Brownian CPP models.

Contribution

It establishes the convergence of the genealogy conditioned on recent MRCA to a Brownian CPP and characterizes the distribution of coalescence times in this limit.

Findings

Genealogy converges to Brownian CPP as MRCA becomes recent.

Coalescence times become i.i.d. uniform in (0,1) in the limit.

Provides a coupling between Kingman coalescent and Brownian CPP models.

Abstract

We consider a continuous population whose dynamics is described by the standard stationary Fleming-Viot process, so that the genealogy of $n$ uniformly sampled individuals is distributed as the Kingman $n$-coalescent. In this note, we study some genealogical properties of this population when the sample is conditioned to fall entirely into a subpopulation with most recent common ancestor (MRCA) shorter than $\varepsilon$. First, using the comb representation of the total genealogy (Lambert & Uribe Bravo 2016), we show that the genealogy of the descendance of the MRCA of the sample on the timescale $\varepsilon$ converges as $\varepsilon\to 0$. The limit is the so-called Brownian coalescent point process (CPP) stopped at an independent Gamma random variable with parameter $n$, which can be seen as the genealogy at a large time of the total population of a rescaled critical birth-death process, biased by the $n$-th power of its size. Secondly, we show that in this limit the coalescence times of the $n$ sampled individuals are i.i.d. uniform random variables in $(0,1)$. These results provide a coupling between two standard models for the genealogy of a random exchangeable population: the Kingman coalescent and the Brownian CPP.