Annihilation and coalescence on binary trees

TL;DR

This paper analyzes the long-term distribution of infection states at the root of a binary tree under a model of infection spread involving annihilation, coalescence, and mutation, using dynamical systems theory.

Contribution

It provides a complete characterization of the limiting distribution at the root for all initial probabilities and mutation rates, extending understanding of infection dynamics on trees.

Findings

Limiting distribution at the root is characterized for all initial probabilities p.

The model with mutation q is analyzed, revealing how it affects the root distribution.

The dynamics of the infection process are governed by a specific dynamical system.

Abstract

An infection spreads in a binary tree of height n as follows: initially, each leaf is either infected by one of k states or it is not infected at all. The infection state of each leaf is independently distributed according to a probability vector p=(p_1,...,p_{k+1}). The remaining nodes become infected or not via annihilation and coalescence: nodes whose two children have the same state (infected or not) are infected (or not) by this state; nodes whose two children have different states are not infected; nodes whose only one of the children is infected are infected by this state. In this note we characterize, for every p, the limiting distribution at the root node of the tree as the height n goes to infinity. We also consider a variant of the model when k=2 and a mutation can happen, with a fixed probability q, at each infection step. We characterize, in terms of p and q, the limiting distribution at the root node of the tree as the height n goes to infinity. The distribution at the root node is driven by a dynamical system, and the proofs rely on the analysis of this dynamics.