发表机构
University of Warwick（华威大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文证明了定向动力学约束模型在谱隙条件下具有线性时间预截断，解决了遍历区间的猜想，方法结合谱估计与图形耦合。
AI 中文摘要
我们在边长为 $n$ 的 $d$ 维盒子上的定向动力学约束模型（KCMs）中建立了线性时间的预截断（precutoff）性质，前提是有限体积谱隙在 $n$ 上一致地远离零。一个典型例子是东北模型（North-East model），这是一个定义在二维整数格上的 $0$-$1$ 自旋系统，其演化规则如下：每当一个格点的南邻和西邻自旋均为 $0$ 时，该格点以速率 $1$ 通过抛掷一枚 $(1-q)$ 硬币来重置自身自旋；在其他所有时刻，其自旋保持冻结。这在定向情形下解决了一个猜想，该猜想断言：在（开放的）遍历参数区间（$q>q_c$）内，所有 KCMs 均具有线性时间的预截断性质。该结果最近已在一般更新族（general update families）的微扰区间（$q$ 接近 $1$）内得到证明。此前在完整遍历区间内，对于定向更新族（oriented update families）的混合时间的一般上界为 $O(n \log n)$。证明方法结合了某个杀死过程（killed process）的谱估计、与平稳过程的图形耦合（graphical coupling）以及反向见证路径（backward witness path）论证，利用约束的定向性来控制未能耦合的概率。
英文摘要
We establish linear time precutoff for oriented kinetically constrained models (KCMs) on a $d$-dimensional box of side length $n$, whenever the finite-volume spectral gap is bounded away from zero uniformly in $n$. A typical example is the North-East model, a $0$-$1$ spin system on the two-dimensional integer lattice that evolves according to the following rule: whenever a site's southerly and westerly nearest neighbours have spin $0$, with rate one it resets its own spin by tossing a $(1-q)$-coin; at all other times its spin remains frozen. This settles, in the oriented case, a conjecture which states that there is linear time precutoff for all KCMs in the (open) ergodic parameter regime ($q>q_c$). The result has been shown recently, for general update families, in a perturbative regime ($q$ close to $1$). The previous general upper bound on the mixing time for oriented update families, in the full ergodic regime, was $O(n \log n)$. The proof combines a spectral estimate for a certain killed process with a graphical coupling to a stationary process and a backward witness path argument, exploiting the orientation of the constraints, to control the probability of failing to couple.
Comments11 pages
