Stein方法与一个三次平均场模型
Stein's method and a cubic mean-field model
AI总结：
本文针对含三体和两体相互作用的三次平均场自旋模型，采用Stein可交换对方法证明标度磁化强度的中心极限定理，还推导了相关概率不等式与偏差原理，并得到临界点附近收敛的新极限分布与阈值。
摘要：
本文研究了一种同时包含三体和两体相互作用的平均场自旋模型。在Contucci、Mingione与Osabutey近期的论文中，已证明大体积下的平衡测度存在三个纯态：两个磁化方向相反的态，以及一个磁化强度为零的非极化态，三者在临界点处合并，且作者还证明了经适当标度的磁化强度满足中心极限定理。本文的研究目标是采用Stein提出的可交换对方法，证明标度后磁化强度的中心极限定理。此外，我们还证明了经适当标度的磁化强度的（非一致）Berry-Esseen界、一个集中不等式、Cramér型中偏差以及中偏差原理。值得注意的是，我们分析了当模型参数$(K_n,J_n)$沿不同斜率的直线以特定速度收敛到临界点$(0,1)$时的Berry-Esseen界，得到了新的极限分布以及收敛速度的阈值。
英文摘要：
In this paper, we study a mean-field spin model with three- and two-body interactions. In a recent paper by Contucci, Mingione and Osabutey, the equilibrium measure for large volumes was shown to have three pure states, two with opposite magnetization and an unpolarized one with zero magnetization, merging at the critical point. The authors proved a central limit theorem for the suitably rescaled magnetization. The aim of our paper is presenting a prove of a central limit theorem for the rescaled magnetization applying the exchangeable pair approach due to Stein. Moreover we prove (non-uniform) Berry-Esseen bounds, a concentration inequality, Cramér-type moderate deviations and a moderate deviations principle for the suitably rescaled magnetization. Interestingly we analyze Berry-Esseen bounds in case the model-parameters $(K_n,J_n)$ converge to the critical point $(0,1)$ on lines with different slopes and with a certain speed, and obtain new limiting distributions and thresholds for the speed of convergence.
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