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

平均场自旋玻璃模型中吉布斯测度的稳定性和结构

• 批准号: 1205781
• 负责人: Dmitriy Panchenko
• 金额: $ 15.02万
• 依托单位: Texas A&M University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2012
• 资助国家: 美国
• 起止时间: 2012-07-01 至 2015-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1205781&HistoricalAwards=false
• 关键词: Stability; Structure; Gibbs; Measures; Mean

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.

我们建议在几种平均场自旋玻璃模型中研究Gibbs测量的一些已知稳定性，并为其他模型开发新的稳定性，并探索可以从其稳定性中推导有关Gibbs测量的哪些类型的信息。任何给定模型中最重要的目标之一是了解其Gibbs测量的渐近结构，在许多模型中，该结构有望用Parisi Ultrabetric Ansatz的版本来描述。 对于稀释的模型，例如稀释的P-SAT和P-Spin模型，该提案涉及对稀释的巴黎ANSATZ的框架的更好的数学理解，该框架是通过随机度量对可测量功能的随机度量描述的，尤其是在这些模型中使用一些最新稳定性结果的新方法。对于PercePtron类型模型，该建议旨在开发新的稳定性属性，并应用于某些腔计算。旋转眼镜一般区域​​中的许多模型源于试图了解不同科学分支（物理学，计算机科学，生物学）的各种优化问题的行为，更具体地说是它们的平均或典型行为，而不是专注于一种固定的情况。这是通过随机化问题参数，然后尝试使用统计物理学和概率理论的方法来回答几个关键问题的方法来完成的。在七十年代和八十年代，物理学家开发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Sherrington-Kirkpatrick模型的环境中，然后随后将这些想法成功地应用于其他模型。物理学家的思想在很大程度上是启发式方法，并且经常用德语单词“ ansatz”描述，意思是“受过教育的猜测，后来通过其结果验证”。近年来，在Sherrington-Kirkpatrick模型的环境中，尤其是严格确认了许多这些想法。该项目的目的是建立在最近的进步基础上，并试图确认对物理学家的其他预测，这些预测对于更广泛的思想适用性至关重要。