CUTOFF PHENOMENON IN NONLINEAR RECOMBINATIONS

• 成果类型: Article
• 署名作者: Caputo, Pietro; Labbe, Cyril; Lacoin, Hubert
• 署名单位: Roma Tre University; Universite Paris Cite; Instituto Nacional de Matematica Pura e Aplicada (IMPA)
• 刊物名称: ANNALS OF APPLIED PROBABILITY
• ISSN/ISSBN: 1050-5164
• DOI: 10.1214/24-AAP2139
• 发表日期: 2025
• 页码: 1164-1197
• 关键词: equation; speed; time

摘要

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.