Nonlinear recombinations and generalized random transpositions

TL;DR

This paper analyzes a nonlinear recombination model from population genetics, approximating it with a mean field linear evolution via generalized random transpositions, and provides quantitative bounds on chaos propagation and convergence rates.

Contribution

It introduces a novel approach linking nonlinear recombination models to generalized random transpositions, with rigorous bounds on chaos propagation and convergence speed.

Findings

Uniform in time propagation of chaos with quantitative bounds

Entropy production estimate for generalized random transpositions

Sharp convergence rates to stationarity in entropy and total variation

Abstract

We study a nonlinear recombination model from population genetics as a combinatorial version of the Kac-Boltzmann equation from kinetic theory. Following Kac's approach, the nonlinear model is approximated by a mean field linear evolution with a large number of particles. In our setting, the latter takes the form of a generalized random transposition dynamics. Our main results establish a uniform in time propagation of chaos with quantitative bounds, and a tight entropy production estimate for the generalized random transpositions, which holds uniformly in the number of particles. As a byproduct of our analysis we obtain sharp estimates on the speed of convergence to stationarity for the nonlinear equation, both in terms of relative entropy and total variation norm.