Structure of the Gibbs distribution in spin glass models with applications

这个提议中研究的主要问题来自物理学家大约40年前引入的模型，目的是理解某些金属合金（称为自旋玻璃）的不寻常磁性行为。对这些模型的研究产生了许多新的想法，这些想法超出了它们最初的动机，找到了意想不到的应用。 最有趣的应用之一是优化问题领域。举一个例子，让我们考虑一个很大的群体，其中任何两个人要么是朋友，要么是敌人，让我们把他们分成两组，试图保持朋友在一起，分开敌人。这并不总是能完美地做到，因为如果你的两个朋友是敌人，那么要么你失去了一个朋友，要么他们会在你的群体中继续成为敌人;这些被称为沮丧的三重人格。这个以最佳方式划分群体的问题通常很难解决（想象一个有许多失败的三元组的情况），但是人们可以问很多有趣的问题，关于在典型情况下会发生什么。例如，人们可以问最优解通常是什么样子的，或者其他几乎最优的解与最优解有什么关系。 物理学家们非常成功地将自旋玻璃研究中提出的思想应用于数学（旅行商问题、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，他们已经开发了一个非常深刻和复杂的理论图片描述在这些优化问题的典型情况下发生了什么（其中典型情况是通过随机选择问题的参数来建模的）。另一方面，考虑到这一点，他们可以提出有效的算法来解决实际中的一些问题。 该提案的重点是第一类，即物理学家在一系列模型中的理论图景--所谓的稀释自旋玻璃模型，包括计算机科学中的随机可满足性模型或上述问题的修改。在每个人只与少数其他人互动的情况下，将一群人分成两组。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严格的数学解释。特别是，该提案试图证明一个著名的公式的自由能和描述的吉布斯措施在这些模型的结构。就像物理学家的工作是基于在自旋玻璃模型的背景下发展的想法一样，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方面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.