Cutoff phenomenon in nonlinear recombinations
Fuente:
arXiv
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| Autores principales: | , , |
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| Formato: | Preprint |
| Publicado: |
2024
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| Materias: | |
| Acceso en línea: | |
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| _version_ | 1866929526068477952 |
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| author | Caputo, Pietro Labbé, Cyril Lacoin, Hubert |
| author_facet | Caputo, Pietro Labbé, Cyril Lacoin, Hubert |
| contents | We investigate a quadratic dynamical system known as nonlinear recombinations. This system models the evolution of a probability measure over the Boolean cube, converging to the stationary state obtained as the product of the initial marginals. Our main result reveals a cutoff phenomenon for the total variation distance in both discrete and continuous time. Additionally, we derive the explicit cutoff profiles in the case of monochromatic initial distributions. These profiles are different in the discrete and continuous time settings. The proof leverages a pathwise representation of the solution in terms of a fragmentation process associated to a binary tree. In continuous time, the underlying binary tree is given by a branching random process, thus requiring a more elaborate probabilistic analysis. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2402_11396 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Cutoff phenomenon in nonlinear recombinations Caputo, Pietro Labbé, Cyril Lacoin, Hubert Probability 82C20, 60K35, 60J80 We investigate a quadratic dynamical system known as nonlinear recombinations. This system models the evolution of a probability measure over the Boolean cube, converging to the stationary state obtained as the product of the initial marginals. Our main result reveals a cutoff phenomenon for the total variation distance in both discrete and continuous time. Additionally, we derive the explicit cutoff profiles in the case of monochromatic initial distributions. These profiles are different in the discrete and continuous time settings. The proof leverages a pathwise representation of the solution in terms of a fragmentation process associated to a binary tree. In continuous time, the underlying binary tree is given by a branching random process, thus requiring a more elaborate probabilistic analysis. |
| title | Cutoff phenomenon in nonlinear recombinations |
| topic | Probability 82C20, 60K35, 60J80 |
| url | https://arxiv.org/abs/2402.11396 |