Annihilation and coalescence on binary trees

An infection spreads in a binary tree $\mathcal{T}-n$ 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 = (p₁, ..., 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 such that only one of the children is infected are infected by this state. In this paper we characterize, for every p, the limiting distribution at the root node of $\mathcal{T}-n$ as 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 $\mathcal{T}-n$ as 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.