平均场铁磁 Potts 模型系统扫描动力学的混合与截止
Mixing and cutoff for the systematic scan dynamics of the mean-field ferromagnetic Potts model
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中文总结 AI 辅助
研究平均场铁磁 Potts 模型系统扫描动力学的混合时间，证明对于特定条件，其混合时间为 c(β,q)log n + Θ(1)，呈现截止现象，这是自旋系统中该动力学的首个一般截止结果，对马尔可夫链理论有独立意义。
中文摘要 AI 辅助
我们研究了 n 顶点完全图上 q 态铁磁 Potts 模型（即平均场模型）的系统扫描动力学的混合时间。该马尔可夫链按固定预定顺序依次更新顶点，与每次更新均匀随机顶点的 Glauber 动力学不同。系统扫描动力学在实践中很有吸引力，但其理论分析远不如 Glauber 动力学发达。我们表明，对于每个 q≥2 和 β<βs（βs 是与 Glauber 动力学慢混合开始相关的亚稳阈值），铁磁平均场 Potts 模型的系统扫描动力学在 Θ(log n)次扫描或等效地在 Θ(n log n)次单站点更新中混合。事实上，我们证明了一个更精确的结果，即存在常数 c(β,q)>0，使得混合时间为 c(β,q)log n + Θ(1)，这意味着马尔可夫链表现出截止现象，总变差距离在狭窄的 Θ(1)时间窗口内从接近 1 急剧下降到接近 0。该结果在 β 方面也是严格紧的，因为对于 β>βs，动力学混合指数缓慢。据我们所知，这是自旋系统中系统扫描动力学的第一个一般截止结果。该结果在马尔可夫链理论中也可能具有独立的意义，因为系统扫描动力学既是全局的又是不可逆的，这两种情况下截止仍然知之甚少。
英文摘要
We study the mixing time of the systematic scan dynamics for the $q$-state ferromagnetic Potts model on the $n$-vertex complete graph, known as the mean-field model. This Markov chain updates vertices sequentially according to a fixed predetermined order, in contrast to the Glauber dynamics which updates a uniformly random vertex at each step. Systematic scan dynamics are attractive in practice as they often demonstrate strong empirical performance. However, their theoretical analysis remains far less developed than that of the Glauber dynamics. We take a step toward addressing this imbalance by showing that for every $q\ge 2$ and $β<β_s$, where $β_s$ is the metastability threshold associated with the onset of slow mixing for the Glauber dynamics, the systematic scan dynamics for the ferromagnetic mean-field Potts model mixes in $Θ(\log n)$ scans or, equivalently, in $Θ(n\log n)$ single site updates. We in fact prove a sharper result; namely, that there exists a constant $c(β,q) > 0$ such that the mixing time is $c(β,q)\log n + Θ(1),$ which implies that the Markov chain exhibits the cutoff phenomenon, with the total variation distance to the stationary distribution dropping abruptly from nearly 1 to nearly 0 within a narrow $Θ(1)$ time window. This result is tight in $β$ as well since the dynamics mixes exponentially slowly for $β> β_s$. To the best of our knowledge, this is the first general cutoff result for the systematic scan dynamics in the context of spin systems. The result may also be of independent interest in the theory of Markov chains, since the systematic scan dynamics is both global and non-reversible, two settings in which cutoff remains poorly understood.
