Horton self-similarity of Kingman's coalescent tree

TL;DR

This paper proves a weak form of Horton self-similarity in Kingman's coalescent trees using differential equations and explores its connection to white noise level set trees, revealing structural similarities.

Contribution

It introduces a weak Horton self-similarity result for Kingman's coalescent trees and links this property to white noise level set trees.

Findings

Establishes a weak Horton self-similarity for Kingman's coalescent trees.

Connects Kingman's trees to white noise level set trees, implying similar self-similarity.

Uses differential equations to analyze the structure of coalescent trees.

Abstract

The paper establishes a weak version of Horton self-similarity for a tree representation of Kingman's coalescent process. The proof is based on a Smoluchowski-type system of ordinary differential equations for the number of branches of a given Horton-Strahler order in a tree that represents Kingman's N-coalescent process with a constant kernel, in a hydrodynamic limit. We also demonstrate a close connection between the combinatorial Kingman's tree and the combinatorial level set tree of a white noise, which implies Horton self-similarity for the latter.