CAREER: Stochastic processes in statistical physics and optimization

随机过程在许多物理学科中起着重要的作用。目前的建议侧重于统计物理和组合优化中自然产生的那些过程。提出的问题的共同特点是公式简单，基本的数学结构，有趣的潜在现象和对物理学科的重大影响。该建议的一个主要方面是关于高斯过程的极值。一个例子是关于一些空间过程的水平集的几何，特别是一个人是否可以在一个随机的表面上行走，而大部分时间都停留在高山上。另一个主要方面是关于随机约束满足问题的相变问题，其中一个例子是判定是否存在一个赋值同时满足一组随机布尔公式。此外，PI打算将概率论应用于统计学习和生物进化等相关领域。例如，PI希望了解个人的特征如何影响社会网络的结构，以及从网络结构中可以了解到关于个人的什么。最后，PI打算为概率论的研究生提供研究机会，并开发主题课程，将概率论技术带给相关领域的学生。本提案的主题是在统计物理和优化中的一些随机过程的新理论和应用的发展。在高斯过程方向上，重点研究了二维高斯自由场水平集的几何形状、对极大测度理论的改进以及高斯自由场与随机漫步之间的联系。例如，我们打算研究与Liouville量子引力有关的二维高斯自由场上的随机几何和随机运动。在随机csp和优化问题的方向上，该提议的特点是统计物理学家预测的解空间的有趣相变。由于大多数经典np完全问题都表示为csp，而随机csp是计算困难的csp的丰富来源，因此对随机csp的研究有望揭示算法性能的潜在障碍。随机组合优化问题的一些研究与理解某些广泛使用的算法的平均复杂度有关。此外，PI建议研究社会网络的某些概率模型，如随机几何图，以及生物进化中的nk适应度模型，目的是为一些实验结果提供数学解释。

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.