任意有限乘积空间中非均匀非线性重组的截断现象
Cutoff Phenomenon for Inhomogeneous Nonlinear Recombination in Arbitrary Finite Product Spaces
AI总结：
本文研究离散时间非线性重组模型，通过开发密度涨落的新颖代数表示，将截断现象推广至有限自旋和非均匀平稳测度的一般情形。
摘要：
在本文中，我们证明了一般离散时间非线性重组模型的截断现象。该系统建模了有限乘积空间 $S^n$ 上概率测度的演化，表示 $n$ 个位置上的自旋状态。尽管其平稳分布具有乘积结构，且其演化是马尔可夫的，但该模型的动力学是非线性的。因此，混合时间的估计成为一项高度非平凡的任务。Caputo, Labbé 和 Lacoin [The Annals of Applied Probability 35:1164-1197, 2025] 考虑了具有两种自旋和均匀平稳测度的特殊情况，其中已验证了混合行为的截断现象。在本文中，我们通过为系统相对于其平稳状态的密度涨落开发一种新颖的代数表示，将此结果推广到具有有限自旋和非均匀平稳测度的一般情形。
英文摘要：
In this article, we prove the cutoff phenomenon for a general class of the discrete-time nonlinear recombination models. This system models the evolution of a probability measure on a finite product space $S^n$ representing the state of spins on $n$ sites. Although its stationary distribution has a product structure, and its evolution is Markovian, the dynamics of the model is nonlinear. Consequently, the estimation of the mixing time becomes a highly non-trivial task. The special case with two spins and homogeneous stationary measure was considered in Caputo, Labbé, and Lacoin [The Annals of Applied Probability 35:1164-1197, 2025], where the cutoff phenomenon for the mixing behavior has been verified. In this article, we extend this result to the general case with finite spins and inhomogeneous stationary measure by developing a novel algebraic representation for the density fluctuation of the system with respect to its stationary state.
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