Fragmentation process, pruning poset for rooted forests, and M\"obius inversion

TL;DR

This paper models a fragmentation process relevant to population genetics, using a pruning poset and Möbius inversion to analyze the probabilities of rooted forests representing the process's realizations.

Contribution

It introduces a novel pruning poset framework and applies Möbius inversion to compute probabilities in a complex fragmentation process.

Findings

Constructed an auxiliary i.i.d. process for pathwise realization

Derived explicit probability expressions for rooted forests

Connected fragmentation dynamics with combinatorial poset structures

Abstract

We consider a discrete-time Markov chain, called fragmentation process, that describes a specific way of successively removing objects from a linear arrangement. The process arises in population genetics and describes the ancestry of the genetic material of individuals in a population experiencing recombination. We aim at the law of the process over time. To this end, we investigate sets of realisations of this process that agree with respect to a specific order of events and represent each such set by a rooted (binary) tree. The probability of each tree is, in turn, obtained by M\"obius inversion on a suitable poset of all rooted forests that can be obtained from the tree by edge deletion; we call this poset the \textit{pruning poset}. Dependencies within the fragments make it difficult to obtain explicit expressions for the probabilities of the trees. We therefore construct an auxiliary process for every given tree, which is i.i.d. over time, and which allows to give a pathwise construction of realisations that match the tree.