Stability and Structure of Gibbs' Measures in Mean-field Spin Glass Models

我们建议研究几种平均场自旋玻璃模型中吉布斯测度的一些已知稳定性特性，并为其他模型开发新的稳定性特性，以及探索可以从其稳定性特性中推断出有关吉布斯测度的哪些信息。任何给定模型中最重要的目标之一是了解其吉布斯测度的渐近结构，并且在许多模型中，该结构预计由 Parisi ultrametric Ansatz 的一个版本来描述。 对于稀释模型，例如稀释 p-sat 和 p-spin 模型，该提案涉及对通过可测量函数空间上的随机测量描述的稀释 Parisi ansatz 框架进行更好的数学理解，特别是寻找新方法来利用这些模型中的一些最新稳定性结果。对于感知器类型模型，该提案旨在开发新的稳定性属性，并将其应用于某些腔计算。自旋玻璃一般领域中的许多模型都源于尝试了解不同科学分支（物理学、计算机科学、生物学）的各种优化问题的行为，更具体地说，了解它们的平均或典型行为，而不是专注于一种固定场景。这是通过随机化问题的参数，然后尝试使用统计物理学和概率论的方法回答几个关键问题来完成的。在七八十年代，物理学家提出了许多新颖的想法来解决这些非常困难的问题，首先是在现在著名的谢林顿-柯克帕特里克模型的背景下，然后又成功地将这些想法应用到其他模型中。物理学家的想法在很大程度上是启发式的，通常用德语单词“Ansatz”来描述，意思是“有根据的猜测，后来通过其结果得到验证”。近年来，其中许多想法都得到了严格的证实，特别是在谢林顿-柯克帕特里克模型的背景下。该项目的目标是在最近取得的进展的基础上，尝试证实物理学家的其他甚至更大胆的预测，这些预测对于更广泛地应用他们的想法至关重要。

We propose to study some known stability properties of the Gibbs measures in several mean-field spin glass models and to develop new stability properties for other models, as well as to explore what kind of information about the Gibbs measures can be deduced from their stability properties. One of the most important objectives in any given model is to understand the asymptotic structure of its Gibbs measure and in a number of models this structure is expected to be described by a version of the Parisi ultrametric Ansatz. For diluted models, such as diluted p-sat and p-spin models, the proposal concerns with a better mathematical understanding of a framework for the diluted Parisi ansatz described by a random measure on the space of measurable functions and, in particular, with finding new ways to utilize some recent stability results in these models. For perceptron type models, the proposal aims to develop new stability properties with applications to certain cavity computations. Many models in the general area of Spin Glasses originate from the attempts to understand the behavior of various optimization problems from different branches of science (physics, computer science, biology) and, more specifically, their average or typical behavior rather than focusing on one fixed scenario. This is done by randomizing the parameters of the problem and then trying to answer several key questions using the methods from Statistical Physics and Probability Theory. In the seventies and eighties, the physicists developed a number of novel ideas to approach these very difficult questions, first, in the setting of the now famous Sherrington-Kirkpatrick model, and then later successfully applied these ideas to other models as well. The ideas of the physicists were for the most part heuristic and are often described by the German word "Ansatz" which means "an educated guess that is verified later by its results". In recent years, many of these ideas have been confirmed rigorously, especially, in the setting of the Sherrington-Kirkpatrick model. The goal of this project is to build upon recent progress and try to confirm other, even more bold, predictions of the physicists that are crucial for broader applicability of their ideas.