Nonlinear recombinations and generalized random transpositions

Pietro Caputo; Daniel Parisi

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arXiv · 2207.04775

Nonlinear recombinations and generalized random transpositions

Pietro Caputo·Daniel Parisi

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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.

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BibTeX RIS

Pietro Caputo, Daniel Parisi. 2022-07-11. Nonlinear recombinations and generalized random transpositions. https://arxiv.org/abs/2207.04775

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