Weak limits for the largest subpopulations in Yule processes with high mutation probabilities

TL;DR

This paper analyzes the behavior of large subpopulations in Yule processes with high mutation probabilities, establishing Poisson limit laws and applying results to percolation on recursive trees.

Contribution

It introduces a general method for deriving Poisson limits for large subpopulations in Yule processes with high mutation rates and applies it to specific mutation regimes and percolation models.

Findings

Poisson limit laws for subpopulation sizes are established.

The method applies to various mutation regimes of interest.

Results are extended to subcritical Bernoulli bond percolation on recursive trees.

Abstract

We consider a Yule process until the total population reaches size $n\gg 1$, and assume that neutral mutations occur with high probability $1-p$ (in the sense that each child is a new mutant with probability $1-p$, independently of the other children), where $p=p_n\ll 1$. We establish a general strategy for obtaining Poisson limit laws for the number of subpopulations exceeding a given size and apply this to some mutation regimes of particular interest. Finally, we give an application to subcritical Bernoulli bond percolation on random recursive trees with percolation parameter $p_n$ tending to zero.