Structure of the Gibbs distribution in spin glass models with applications

本提案中研究的主要问题源于大约40年前物理学家提出的模型，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。******最有趣的应用之一是优化问题的领域。举个例子，让我们考虑一大群人，其中任意两个人要么是朋友，要么是敌人，让我们把他们分成两组，试图保持朋友在一起，把敌人分开。这并不总是能够做到完美，因为如果你的两个朋友是敌人，那么你要么失去其中一个朋友，要么他们将继续在你的团队中成为敌人；这些被称为受挫三元组。一般来说，以最优方式划分团队的问题很难解决（想象一个有许多令人沮丧的三元组的情况），但在典型情况下，人们可以提出许多有趣的问题。例如，人们可以问最优解通常是什么样的，或者其他几乎最优解与最优解有什么关系。******物理学家非常成功地将自旋玻璃研究中发展出来的思想应用于数学（旅行推销员问题、图划分）、计算机科学（布尔公式的随机可满足性）和生物学（大脑建模）中出现的各种优化问题，他们的主要贡献可以大致分为两类。一方面，他们发展了一个非常深刻和复杂的理论图景，描述了这些优化问题（通过随机选择问题的参数来建模典型情况）中典型情况下发生的情况。另一方面，有了这幅图，他们就能在实践中想出解决这些问题的有效算法。******本提案的重点是第一类，即物理学家在一个模型家族中的理论图景-所谓的稀释自旋玻璃模型，包括计算机科学中的随机满意度模型或上述问题的修改，即在每个人只与少数其他人互动的情况下将一组人分成两组。该提案的主要目标是为物理学家在这些模型中提出的图像找到一个严格的数学解释。特别是，该提案试图证明一个著名的自由能公式，并描述这些模型中吉布斯测度的结构。正如物理学家的工作是建立在自旋玻璃模型的背景下发展起来的，这个提议建立在近年来在原始自旋玻璃模型背后的严格数学理论取得的重大进展之上。

The main questions studied 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 behavior 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 (brain modeling), and their main contributions can be roughly divided into two categories. On the one hand, they have developed a very deep and sophisticated theoretical picture describing what happens in a typical situation in these optimization problems (where a typical situation is modeled 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 focus of this proposal is on the first category, namely, the physicists' theoretical picture in one family of models - the so called diluted spin glass models that include the random satisfiability models from computer science or 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 a famous formula for the free energy and describe the structure of the Gibbs measure in these models. 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.