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.
" have been successfully solved within the scheduled due date. The first research problem in the project proposal was the problem of proving "chaos" in certain spin glass systems. Spin glasses are certain kinds of magnetic materials. It was conjectured by physicists more than twenty years ago that spin glasses are "chaotic" in nature, in the sense that they exhibit sudden and dramatic shifts in response to small changes in the environment. One of the major outcomes of the project is a mathematical proof of chaos in certain kinds of spin glasses. Additionally, the PI managed to connect the phenomenon of chaos with two other features of spin glasses: multiple valleys and superconcentration. The second problem posed in the project proposal was the problem of computing the large deviation rate function for dense random graphs. The theory of large deviations studies the effect of the occurrence of rare events. The PI and the collaborators were the first to develop a systematic theory of large deviations for random graphs. This was the second major outcome of the project. Besides the two major outcomes described above, the PI and his collaborators also solved two other problems stated in the proposal, namely, the problem of understanding the limiting behavior of random graphs with a given degree sequence, and the limit behavior of certain objects arising from number theory. The broader impacts of these outcomes are many. Spin glasses are important objects in physics, and the mathematical proof of chaos and multiple valleys in spin glasses confirms previously unconfirmed physical conjectures. Random graphs have become increasingly useful for the study of large networks such as the internet, and it is important to understand the effect of rare events occurring in such networks. The large deviation theory developed in this project has already been useful to practitioners: it has found applications in the theory of the so-called "Exponential Random Graph Models" that are widely used by statisticians and social scientists.
