Stochastic process plays a fundamental role in a number of physical disciplines. The current proposal focuses on those processes that arise naturally in statistical physics and combinatorial optimization. The common features of the proposed problems are simple formulation, fundamental mathematical structure, interesting underlying phenomena and non-trivial impacts on physical disciplines. One main aspect of the proposal is on extreme values for Gaussian processes. An example question is on the geometry of level sets for some spatial processes, and in particular whether one could walk on a random surface while staying on high mountains most of the time. Another main aspect is on phase transitions of random constraint satisfaction problems, and an example question is to decide whether there exists an assignment simultaneously satisfying a collection of random boolean formulae. In addition, the PI intends to apply probability in related areas such as statistical learning and biological evolution. For example, the PI wishes to understand how features of individuals influence the structure of social network and what could be learned about individuals from the network structure. Finally, the PI intends to provide research opportunities for both graduate students in probability theory, and to develop topic courses that bring probability techniques to students in related areas.
The main theme of this proposal is the development of new theory and applications on a number of stochastic processes in statistical physics and optimization. In the direction of Gaussian processes, the proposal focuses on a number of aspects including the geometry of level sets for two-dimensional Gaussian free fields, an improvement on majorizing measure theory, as well as the connection between Gaussian free fields and random walks. For instance, we intend to study the random geometry and random motion on the two-dimensional Gaussian free field, which is connected to the Liouville quantum gravity. In the direction of random CSPs and optimization problems, the proposal features the intriguing phase transitions of the solution spaces predicted by statistical physicists. Since most classical NP-complete problems are expressed as CSPs and random CSPs are a rich source of computationally hard CSPs, the proposed study of random CSPs are expected to shed light on underlying barriers to algorithmic performance. Some of the study of random combinatorial optimization problems is related to understanding the average complexity of certain widely-used algorithms. Furthermore, the PI proposes to study certain probabilistic models for social network such as random geometric graphs, as well as the NK-fitness model in biological evolution with the aim of providing mathematical explanation to some experimental findings.
