Variational representations for the Parisi functional and the two-dimensional Guerra-Talagrand bound
Variational representations for the Parisi functional and the two-dimensional Guerra-Talagrand bound
复制标题
Parisi 泛函和二维 Guerra-Talagrand 界的变分表示
DOI：
10.1214/16-aop1154
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发表时间：
2015
期刊：
影响因子：
--
通讯作者：
Wei
中科院分区：
文献类型：
--
作者：
Wei
Sherrington-Kirkpatrick模型（SK）中Parisi公式的有效性最初由Talagrand [18]证明。其中的中心论点依赖于通过二维Guerra-Talagrand（GT）副本对称性破缺束缚对耦合自由能的非常专门的研究。人们认为这个界及其高维推广与SK模型中的温度混沌和超度量性密切相关，但完整的研究仍然很难实现。受Bovier-Klimovsky [2]和Auffinger-Chen [3]的启发，本文的目的是提出一种新的方法来分析最优随机控制问题中混合p$-自旋模型的Parisi泛函和二维GT界.我们计算的Parisi功能的方向导数和导出的Parisi措施的等价准则。我们展示了我们的方法如何为GT界提供一个简单而有效的控制，从而在Talagrand的重叠正性[20，第14.12节]和Chatterjee [5]和Chen [6]中的无序混沌中产生了几个新的结果。特别是，我们提供了一些例子的模型包含奇数$p$自旋相互作用。
The validity of the Parisi formula in the Sherrington-Kirkpatrick model (SK) was initially proved by Talagrand [18]. The central argument therein relied on a very dedicated study of the coupled free energy via the two-dimensional Guerra-Talagrand (GT) replica symmetry breaking bound. It is believed that this bound and its higher dimensional generalization are highly related to the conjectures of temperature chaos and ultrametricity in the SK model, but a complete investigation remains elusive. Motivated by Bovier-Klimovsky [2] and Auffinger-Chen [3], the aim of this paper is to present a novel approach to analyzing the Parisi functional and the two-dimensional GT bound in the mixed $p$-spin models in terms of optimal stochastic control problems. We compute the directional derivative of the Parisi functional and derive equivalent criteria for the Parisi measure. We demonstrate how our approach provides a simple and efficient control for the GT bound that yields several new results on Talagrand's positivity of the overlap [20,Section 14.12] and disorder chaos in Chatterjee [5] and Chen [6]. In particular, we provide some examples of the models containing odd $p$-spin interactions.