Crossing bridges between percolation models and Bienaym\'e-Galton-Watson trees

TL;DR

This survey explores the connections between percolation models and Galton-Watson trees, highlighting new links and constructions that unify concepts in probability theory and population genetics.

Contribution

It introduces a novel connection between Divide-and-Color percolation and multi-type Galton-Watson trees, unifying different probabilistic models.

Findings

Unified construction of Galton-Watson trees via percolation and mutations

New link between Divide-and-Color percolation and multi-type trees

Overview of how these models relate in probability and genetics

Abstract

In this survey, we explore the connections between two areas of probability: percolation theory and population genetic models. Our first goal is to highlight a construction on Galton-Watson trees, which has been described in two different ways: Bernoulli bond percolation and neutral mutations. Next, we introduce a novel connection between the Divide-and-Color percolation model and a particular multi-type Galton-Watson tree. We provide a gentle introduction to these topics while presenting an overview of the results that connect them.