Volume 7 (2024)
Caputo, Pietro;  Parisi, Daniel
Nonlinear recombinations and generalized random transpositions
Annales Henri Lebesgue, Volume 7 (2024), pp. 1245-1299.

Metadata

DOI10.5802/ahl.219
Keywords Nonlinear recombinations, Entropy, Kac program, Permutations, Logarithmic Sobolev inequalities

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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