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Crossing bridges between percolation models and Bienaymé-Galton-Watson trees

Airam Blancas, María Clara Fittipaldi, Saraí Hernández-Torres

TL;DR

This paper builds a cohesive bridge between percolation theory and BGW-type branching processes by showing that Bernoulli percolation on BGW trees corresponds to neutral mutation dynamics with infinite alleles, and that Divide-and-Color percolation extends this link to finite-allele models. It systematically relates percolation constructs (Bernoulli, DaC) to multi-type and mutation-driven BGW processes, clarifying how extinction, survival, and phase transitions translate across the two frameworks. The main contributions include explicit correspondences between percolation parameters and mutation rates, and precise characterizations of phase transitions for BGW trees under various mutation schemes. This cross-disciplinary perspective enhances understanding of large-scale structures in random trees and provides tools for analyzing mutation-driven genealogies through percolation techniques.

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.

Crossing bridges between percolation models and Bienaymé-Galton-Watson trees

TL;DR

This paper builds a cohesive bridge between percolation theory and BGW-type branching processes by showing that Bernoulli percolation on BGW trees corresponds to neutral mutation dynamics with infinite alleles, and that Divide-and-Color percolation extends this link to finite-allele models. It systematically relates percolation constructs (Bernoulli, DaC) to multi-type and mutation-driven BGW processes, clarifying how extinction, survival, and phase transitions translate across the two frameworks. The main contributions include explicit correspondences between percolation parameters and mutation rates, and precise characterizations of phase transitions for BGW trees under various mutation schemes. This cross-disciplinary perspective enhances understanding of large-scale structures in random trees and provides tools for analyzing mutation-driven genealogies through percolation techniques.

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.

Paper Structure

This paper contains 30 sections, 7 theorems, 46 equations, 11 figures.

Table of Contents

  1. Introduction
  2. Trees
  3. Trees as graphs
  4. Trees as genealogical structures
  5. Genealogical structures for branching processes
  6. Bienaymé-Galton-Watson process
  7. Multi-type BGW processes
  8. BGW processes with neutral mutations and infinite alleles
  9. BGW processes with neutral mutations and finite alleles
  10. Mother-dependent mutation model
  11. Mother-independent mutation model
  12. Phase Transition for branching processes
  13. Extinction for BGW processes
  14. Phase transition for BGW process with neutral mutation
  15. A glance at Bernoulli percolation
  16. ...and 15 more sections

Key Result

Proposition 4.1

Let $T_d$ be a $d$-ary tree with $d \geq 2$ and consider Bernoulli percolation with parameter $p \in [0,1]$ on $T_d$. If $\mathcal{T}_{\emptyset}$ is the open cluster of the root, then $\mathcal{T}_{\emptyset}$ follows the distribution of a BGW tree with progeny distribution $\hbox{Bin}(d, p)$.

Figures (11)

  • Figure 1: This figure shows an example of a BGW tree with Ulam-Harris labeling. In this instance, the offspring distribution is defined as $\mu(k)=\tfrac{1}{3} \mathbbm{1} _{ \{1,2,3\} }(k)$.
  • Figure 2: An example of a genealogical tree for our multi-type process is shown. In this case, we label the tree using the usual Ulam-Harris notation, but we include a node color that represents the type of each individual, using blue for type 1, purple for type 2, and green for type 3.
  • Figure 3: An example of a BGW tree with infinite allele-type mutations. This tree corresponds to the one in Figure \ref{['fig1: UH BGW ']} with with $r=0.5$. Note that after each new mutation, a new type of individual appears (a type that has never been seen before).
  • Figure 4: An example of a BGW tree with mother-dependent finite allele mutations. This tree corresponds to the tree in Figure \ref{['fig1: UH BGW ']} with $r=0.5$ and three types represented as circle, square, and pentagon. Note that in this case, each mutation event produces an individual whose type is chosen uniformly among all available types, excluding its mother's type.
  • Figure 5: An illustration of a branching event for the mother-independent mutation (MIM) model in the case where we have three types (circle, square and pentagon). For this model, each mutant child chooses its type uniformly from the set of all possible types.
  • ...and 6 more figures

Theorems & Definitions (8)

  • Proposition 4.1
  • Proposition 4.2
  • Theorem 4.3: Lyons90
  • Proposition 4.4: Haggstrom
  • Proposition 4.5
  • Theorem 5.1: Lyons, 1990
  • Theorem 5.2
  • proof