Rigorous results on Spin Glasses

旋转玻璃的严格结果

• 批准号: RGPIN-2020-07009
• 负责人: Panchenko, Dmitriy
• 金额: $ 1.75万
• 依托单位: University of Toronto
• 依托单位国家: 加拿大
• 项目类别: Discovery Grants Program - Individual
• 财政年份: 2020
• 资助国家: 加拿大
• 起止时间: 2020-01-01 至 2021-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=718021
• 关键词: Rigorous; results; Spin; Glasses

The main questions in this proposal originate from the models that were introduced by the physicists about forty years ago with the goal of understanding the unusual magnetic behaviour of certain metal alloys, called spin glasses. The study of these models produced many new ideas that found unexpected applications beyond their original motivation. One of the most interesting applications was to the area of optimization problems. To give one example, let us consider a large group of people where any two are either friends or enemies, and let us divide them into two groups attempting to keep friends together and separate the enemies. This can not always be done perfectly, because if two of your friends are enemies then either you lose one of your friends or they will stay enemies inside your group; these are called frustrated triples. This problem of dividing the group in an optimal way is known to be very difficult to solve in general (imagine a situation with numerous frustrated triples), but there are many interesting questions one can ask about what happens in a typical situation. For example, one can ask how optimal solutions typically look like, or how other almost optimal solutions are related to the optimal ones. The physicists were very successful in applying the ideas developed in the study of spin glasses to various optimization problems arising in mathematics (traveling salesman problem, graph partitioning), computer science (random satisfiability of Boolean formulas) and biology (modelling brain activity), and their main contributions can be roughly divided into two categories. On the one hand, they have developed a deep and sophisticated theoretical understanding of what happens in a typical situation in these optimization problems (where a typical situation is modelled by choosing the parameters of the problem randomly). On the other hand, having this picture in mind allowed them to come up with efficient algorithms for solving some of these problems in practice. The main focus of this proposal is on the first category, namely, theoretical picture in one family of models -- the so called diluted spin glass models, which includes a modification of the above problem of splitting a group of people into two groups in the case when each person interacts with only a small number of other people. The main goal of the proposal is to find a rigorous mathematical explanation for the picture proposed by the physicists in these models. In particular, the proposal attempts to prove the famous Mézard--Parisi formula for the free energy, as well as describe the structure of the Gibbs measure in these models (where the main open problem is called reproducibility hypothesis). Just like the work of the physicists was based on the ideas developed in the context of the models of spin glasses, this proposal builds upon a significant progress achieved in recent years in the rigorous mathematical theory behind the original spin glass models.

这项提议中的主要问题源于物理学家们在大约40年前提出的模型，目的是理解某些被称为自旋玻璃的金属合金的不寻常的磁性行为。对这些模型的研究产生了许多新的想法，这些想法发现了意想不到的应用，超出了它们最初的动机。