Probability Theory and Spin Glasses.

0243813自旋玻璃的Talagrand平均场模型（如著名的Sherrington柯克帕特里克模型）展示了一些全新的概率论现象，在高水平上可以描述为在相关随机变量的大家族的极值附近存在复杂的结构。对这些模型的严格研究尚处于起步阶段，本提案的主要目标是追求它，其中两个具体目标是获得对p自旋相互作用模型低温阶段的更好理解，并开始对F. Guerra在他的破缺的复制对称中为Sherrington柯克帕特里克模型所束缚。 概率论的中心主题之一是从大量随机事件中出现一种有序。这是由经典定理，如大数定律。这些经典的结果涉及的情况下，随机事件是独立的，也就是说，一个的结果不影响其他的结果。这种假设在实践中是不现实的。我们研究了在大量随机事件中出现的新型集体行为，其中每个事件都可以并且将会影响其他事件的结果。特定的数学模型是由磁化粒子的行为激发的，被称为“自旋玻璃”。

0243813Talagrand Mean field models for spin glasses (such as the famous Sherrington Kirkpatrick model) exhibit some completely new phenomenon of probability theory, that at a high level can be described as the existence of intricate structures among the near extreme values of large families of correlated random variables. The rigorous study of these models is still in its infancy, and the main goal of this proposal is to pursue it. Two of the specific goals are to obtain a better understanding of the low-temperature phase of the p-spin interaction model, and to start the study of the new Hamiltonians introduced by F. Guerra in his broken replica-symmetry bound for the Sherrington Kirkpatrick model. One of the central themes of probability theory is the emergence of a kind of order out of large number of random events. This is expressed by classical theorems, such as the law of large numbers. These classical results relate to situations where the random events are independent, that is, the outcome of one does not influence the outcome of the others. Such an assumption is unrealistic in practice. We investigate the emergence of new types of collective behavior in large collections of random events, where each event can, and will, influence the outcome of the others. The specific mathematical models have been motivated by the behavior of magnetized particles, and are known as "spin glasses".