铁磁伊辛模型的临界与近临界影响界
Critical and near-critical influence bounds for ferromagnetic Ising models
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中文总结 AI 辅助
针对铁磁伊辛模型，证明影响矩阵在临界阈值处有根号n阶界，并推广到近临界情形，进而得出Glauber动力学在超临界窗口内多项式时间混合。
中文摘要 AI 辅助
对于最大度为 $\Delta\ge3$ 的图上的铁磁伊辛模型，我们证明了在树唯一性阈值处，影响矩阵的每一行具有 $\sqrt n$ 量级的界。该估计在度、外场以及所有钉扎条件下是一致的。更一般地，若耦合由 $\beta$ 界定且 $\varepsilon=((\Delta-1)\tanh\beta-1)_+$，则界为 $C(\sqrt n+n\varepsilon)$。证明结合了点态空腔界、正级数磁化倾斜以及 Ding、Song 和 Sun 的场比较定理。该临界估计去除了近期一般图界中铁磁情形下的对数因子。作为推论，零场单点 Glauber 动力学在整个超临界窗口 $\varepsilon=O(\sqrt{\log n/n})$ 内以多项式时间混合，其多项式次数取决于窗口大小。
英文摘要
For a ferromagnetic Ising model on a graph of maximum degree $Δ\ge3$, we prove a bound of order $\sqrt n$ on every row of the influence matrix at the tree uniqueness threshold. The estimate is uniform in the degree, the external fields, and all pinnings. More generally, if the couplings are bounded by $β$ and $\varepsilon=((Δ-1)\tanhβ-1)_+$, the bound is $C(\sqrt n+n\varepsilon)$. The proof combines a pointwise cavity bound with a positive-series magnetization tilt and the field comparison theorem of Ding, Song and Sun. The critical estimate removes the logarithm in recent general graphical bounds for the ferromagnetic case. As a consequence, zero-field single-site Glauber dynamics mixes in polynomial time throughout the supercritical window $\varepsilon=O(\sqrt{\log n/n})$, with the polynomial degree depending on the window size.
