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

我们建议在几个平均场自旋玻璃模型中研究Gibbs测度的一些已知的稳定性性质，并为其他模型发展新的稳定性性质，并探索从它们的稳定性性质可以推导出关于Gibbs测度的什么样的信息。在任何给定的模型中，最重要的目标之一是了解其Gibbs测度的渐近结构，在一些模型中，这种结构有望用Parisi超度量Anatz的一个版本来描述。对于稀疏模型，如稀疏p-sat和p-自旋模型，该提议涉及更好地在数学上理解由可测函数空间上的随机度量描述的稀化Parisi ansatz的框架，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，该建议旨在开发新的稳定性性质，并将其应用于某些空腔计算。自旋眼镜一般领域中的许多模型源于对来自不同科学分支(物理、计算机科学、生物学)的各种优化问题的行为的尝试，更具体地说，是理解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，后来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“Anatz”来描述，意思是“一个有根据的猜测，后来被它的结果证实”。近年来，这些想法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。这个项目的目标是在最新进展的基础上，尝试证实物理学家的其他更大胆的预测，这些预测对于他们的想法的更广泛应用至关重要。

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.