Nonlinear recombinations and generalized random transpositions
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arXiv
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| Autores principales: | , |
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| Formato: | Preprint |
| Publicado: |
2022
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| _version_ | 1866915076301127680 |
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| author | Caputo, Pietro Parisi, Daniel |
| author_facet | Caputo, Pietro Parisi, Daniel |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2207_04775 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Nonlinear recombinations and generalized random transpositions Caputo, Pietro Parisi, Daniel Probability Mathematical Physics Combinatorics 82C40, 82C20, 39B62 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. |
| title | Nonlinear recombinations and generalized random transpositions |
| topic | Probability Mathematical Physics Combinatorics 82C40, 82C20, 39B62 |
| url | https://arxiv.org/abs/2207.04775 |