Cutoff phenomenon in nonlinear recombinations
arxiv(2024)
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.
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