Disordered systems, dense graphs, normal approximation and applications

提出了研究概率论中的几个问题，其中一类问题涉及高维高斯场的结构，并应用于无序系统的研究，包括自旋玻璃和聚合物。第二类问题涉及稠密的随机图，包括未解决的问题的大偏差的子图计数和具有给定的度序列的图形的属性。最后，第三类问题的中心围绕着延续的提议者的早期工作正常的近似在现代问题。在物理系统中，对小扰动的敏感性被广泛地称为混沌。提议者希望研究无序系统中的混沌现象，无序系统是表现出所谓“玻璃”性质的材料的简化物理模型。这项提议的重点是某些被称为自旋玻璃的磁性材料。物理学家和数学家已经研究了近40年，但它们行为的各个方面仍然笼罩在神秘之中，其中之一就是混沌。这位提议者认为，他有新的数学工具来理解这种难以捉摸的现象，他过去已经成功地将其应用于某些聚合物模型。

It is proposed to study several problems in probability.One class of problems concerns the structure of high-dimensional Gaussian fields, with applications to the study of disordered systems, including spin glasses and polymers. A second class of problems involves dense random graphs, including unsolved questions about large deviations of subgraph counts and the properties of graphs with a given degree sequence. Finally, a third class of problems centers around a continuation of the proposer's earlier work of normal approximation in modern problems. Sensitivity to small perturbations in physical systems is broadly known as chaos. The proposer wishes to study the phenomenon of chaos in disordered systems, which are simplified physical models of materials that exhibit so-called `glassy' properties. The key focus of this proposal is on certain kinds of magnetic materials known as spin glasses. These have been studied by physicists and mathematicians for nearly forty years, but various aspects of their behavior are still shrouded in mystery, one of them being chaos. The proposer believes that he has new mathematical tools to understand this elusive phenomenon, which he has already successfully applied to certain polymer models in the past.