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

我们打算研究几种平均场自旋玻璃模型中已知的吉布斯测度的一些稳定性性质，并为其他模型开发新的稳定性性质，并探索从吉布斯测度的稳定性性质中可以推导出哪些关于吉布斯测度的信息。在任何给定的模型中，最重要的目标之一是理解其吉布斯测度的渐近结构，在许多模型中，这种结构预计将由一个版本的Parisi超测量Ansatz来描述。对于稀释模型，如稀释p-sat和p-spin模型，该建议涉及对可测量函数空间上的随机度量所描述的稀释巴黎方差的框架的更好的数学理解，特别是寻找新的方法来利用这些模型中最近的一些稳定性结果。对于感知器类型的模型，该提案旨在开发新的稳定性特性，并将其应用于某些空腔计算。Spin Glasses一般领域的许多模型都源于试图理解来自不同科学分支（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，是它们的平均或典型行为，而不是专注于一个固定的场景。这是通过随机化问题的参数，然后尝试使用统计物理和概率论的方法来回答几个关键问题来完成的。在七八十年代，物理学家们提出了许多新颖的想法来解决这些非常困难的问题，首先，在现在著名的谢林顿-柯克帕特里克模型的背景下，然后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“后来被结果证实的有根据的猜测”。近年来，这些观点中的许多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。这个项目的目标是建立在最近进展的基础上，并试图证实物理学家的其他更大胆的预测，这些预测对他们的想法的更广泛的适用性至关重要。

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.