[Paper Review] A probabilistic analysis of a discrete-time evolution in recombination II. (On partitions)

Servet Martı́nez|arXiv (Cornell University)|Apr 18, 2016

Advanced Topics in Algebra6 references3 citations

TL;DR

This paper develops a probabilistic framework for discrete-time recombination processes in population genetics using general partitions, introducing a Markov chain that models the evolution of genetic composition under recombination. The key contribution is the characterization of quasi-stationary distributions for this chain, showing geometric decay rates and convergence to the limiting product measure, generalizing prior results from dyadic partitions to arbitrary partition families.

ABSTRACT

We study the discrete-time evolution of a transformation on a set of probability measures that is up-dated combining independently the marginals on the atoms of partitions. This model was recently introduced in Baake, Baake and Salamat (Discr. and contin. dynam. syst. 36, 2016) for continuous-time evolution and generalizes previous ones based upon dyadic partitions. We associate to the discrete-time evolution a natural Markov chain and describe its quasi-stationary behavior retrieving all the results we recently found for dyadic partitions.

Motivation & Objective

• To extend the analysis of discrete-time recombination dynamics from dyadic partitions to general families of partitions.
• To model the evolution of genetic composition under recombination as a Markov process on the lattice of partitions.
• To characterize the quasi-stationary behavior of the process before absorption into the limiting product measure.
• To generalize results on geometric decay rates and quasi-stationary distributions from the dyadic case to a broader class of partition structures.

Proposed method

• Define a recombination transformation Ξ that updates a probability measure by combining marginals on atoms of partitions from a given family G.
• Associate the n-th iterate Ξⁿ with a Markov chain (Yₙ) whose state space is the set of partitions in X(G), starting from the coarsest partition.
• Construct a transition matrix P* on the state space, with absorption at the common refinement D(G), and identify the quasi-stationary distribution via a right eigenvector of P* with eigenvalue η.
• Use ratio limit theorems and hitting time analysis to derive the limiting conditional distribution of the chain before absorption.
• Prove that the quasi-stationary distribution is characterized by a transition matrix Q derived from P* and the Perron-Frobenius eigenvector φ.
• Establish that the process exhibits geometric decay rate η < 1, with the survival probability decaying as ηⁿ.

Experimental results

Research questions

• RQ1 How does the discrete-time recombination process behave on general partition families, beyond the dyadic case?
• RQ2 What is the structure of the quasi-stationary distribution for the Markov chain modeling recombination on partitions?
• RQ3 How does the geometric decay rate η relate to the transition dynamics and absorption time?
• RQ4 Can the quasi-stationary behavior be characterized independently of the full absorption process?
• RQ5 What is the role of the common refinement D(G) in determining the limiting behavior of the process?

Key findings

• The Markov chain (Yₙ) associated with the recombination process has a quasi-stationary distribution characterized by a transition matrix Q, which describes the conditional dynamics before absorption.
• The quasi-stationary distribution is supported on the set of partitions ∂(ζ_F), which are those that do not lead to immediate absorption.
• The survival probability decays geometrically at rate η < 1, with limₙ→∞ P(ζ > n) / ηⁿ = constant, confirming exponential decay.
• The transition matrix Q is stochastic and irreducible on the quasi-stationary states, ensuring long-term stability of the conditional distribution.
• The eigenvalue η corresponds to the Perron-Frobenius eigenvalue of the transition matrix P*, and is strictly less than 1.
• The results generalize previous findings on dyadic partitions, with the main technical novelty being the proof of relations (14) and (15) in the general partition case.