局部乘积条件蕴含截止现象
The local product condition implies cutoff
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中文总结 AI 辅助
研究混合时间理论中关于马尔可夫过程截止的猜想，用自然非平衡细化的庞加莱常数γ₊替代原常数，证明混合窗口宽度为O(1/γ₊)，结果通用且证明简洁自洽。
中文摘要 AI 辅助
在混合时间理论中，一个著名的错误猜想预测，一旦马尔可夫过程序列的庞加莱常数与混合时间的乘积发散，该序列就会出现截止现象。我们证明，一旦用其自然的非平衡细化γ₊替代庞加莱常数γ，该陈述就变得正确。更准确地说，我们表明任何马尔可夫过程的混合窗口宽度为O(1/γ₊)。这个估计是精确的，并且在标准正则性假设下是通用的：它适用于有限和无限状态空间，从任何初始条件出发，并且不需要可逆性，也不需要任何形式的链式法则。此外，对于确定性初始化，我们表明γ₊≥κ，其中κ是巴克利 - 埃默里曲率，这使得我们的结果具有广泛的适用性。最后，我们的证明简短且自包含：我们只是遵循用更易于处理的χ²散度代替总变差距离的经典思路，但关键的新颖之处在于参考测度随时间演变，而不是平衡律。
英文摘要
In the theory of mixing times, a famously wrong conjecture predicts that a sequence of Markov processes exhibits cutoff as soon as the product of their Poincaré constant and mixing time diverges. We prove that this statement becomes correct once the Poincaré constant $γ$ is replaced with its natural non-equilibrium refinement, which we denote by $γ_\star$. More precisely, we show that the width of the mixing window of any Markov process is $O(1/γ_\star)$. This estimate is sharp, and universal up to standard regularity assumptions: it holds on finite and infinite state spaces and from any initial condition, and it does not require reversibility, nor any kind of a chain rule. In addition, for deterministic initialization we show that $γ_\star\geκ$, where $κ$ is the Bakry-Émery curvature, making our result broadly applicable. Finally, our proof is short and self-contained: we simply follow the classical idea of replacing the total variation distance by the more tractable $χ^2$-divergence, but with the crucial novelty that the reference measure evolves in time, instead of being the equilibrium law.
