Coalescent processes emerging from large deviations

TL;DR

This paper explores how certain offspring distributions with finite but large variance lead to a new type of coalescent process, extending classical models and connecting to statistical physics phenomena.

Contribution

It introduces a novel coalescent process arising from offspring distributions with stretched exponential tails, bridging evolutionary models and statistical physics.

Findings

Limit coalescent is a β-coalescent.

Connection established with Derrida's REM.

Finite but large variance offspring distributions lead to new coalescent structures.

Abstract

The classical model for the genealogies of a neutrally evolving population in a fixed environment is due to Kingman. Kingman's coalescent process, which produces a binary tree, universally emerges from many microscopic models in which the variance in the number of offspring is finite. It is understood that power-law offspring distributions with infinite variance can result in a very different type of coalescent structure with merging of more than two lineages. Here we investigate the regime where the variance of the offspring distribution is finite but comparable to the population size. This is achieved by studying a model in which the log offspring sizes have a stretched exponential form. Such offspring distributions are motivated by biology, where they emerge from a toy model of growth in a heterogenous environment, but also mathematics and statistical physics, where limit theorems and phase transitions for sums over random exponentials have received considerable attention due to their appearance in the partition function of Derrida's Random Energy Model (REM). We find that the limit coalescent is a $\beta$-coalescent -- a previously studied model emerging from evolutionary dynamics models with heavy-tailed offspring distributions. We also discuss the connection to previous results on the REM.