Large-Scale Phenomena in Models of Statistical Mechanics

PI将研究概率论和统计力学边界上的各种项目。这些方法的一个共同特点是它们都专注于克服技术障碍，这些障碍迄今为止一直阻碍着对潜在物理现象的数学理解。第一组具体的问题研究的随机晶格拉普拉斯算子的频谱特性的作用，在无序介质中的随机游动的标度极限的推导。第二个项目研究引入非吸引势对随机梯度场的静力学和动力学的影响。第三个项目的重点是发展一个详细的方法来本地化的随机游动在一个随机的潜在景观的基础上本征值极端顺序统计格安德森哈密顿。第四个项目利用新的想法来控制统计力学模型中界面的刚度。最后一个项目概述了一种新的方法，基于交换量子数来量化奇异相互作用对相互作用的量子不可区分粒子的大系统的时间演化的影响。该项目将影响我们对各种实际感兴趣的系统的理解，其中均匀化理论，谱分析，微分方程以及概率方法的分析技术，例如，随机分析、极端有序统计和无序系统理论等，发挥重要作用。一些项目的设计，以促进培训，并列入研究，博士后和研究生谁有兴趣在概率论和数学物理。

The PI will investigate a variety of projects on the borderline of probability theory and statistical mechanics. A common feature of these is their focus on overcoming the technical obstacles that have insofar held progress to the mathematical understanding of the underlying physical phenomena. The first set of specific problems studies the role of spectral characteristics of random lattice Laplacians in the derivation of scaling limits of random walks in disordered media. The second project investigates the effects of introducing non-attractive potentials on the statics and dynamics of random gradient fields. The third project focuses on developing a detailed approach to localization of random walks in a random potential landscape based on eigenvalue extreme order statistics for lattice Anderson Hamiltonians. The fourth project utilizes new ideas to control the rigidity of interfaces in statistical mechanical models. The final project outlines a new approach based on exchangeability to quantify the effects of a singular interaction on the time evolution of a large system of interacting quantum indistinguishable particles.The project will impact our understanding of various systems of practical interest where the analytic techniques of homogenization theory, spectral analysis, differential equations as well as probabilistic methods, e.g., stochastic analysis, extreme order statistics and theory of disordered systems, etc., play an important role. A number of projects are devised to facilitate training, and inclusion in research, of postdocs and graduate students who have interest in Probability Theory and Mathematical Physics.