Large-Scale Phenomena in Models of Statistical Mechanics

PI将研究概率论和统计力学边缘的各种项目。这些研究的一个共同特点是，他们致力于克服迄今为止阻碍对潜在物理现象的数学理解取得进展的技术障碍。第一组具体问题研究了随机晶格拉普拉斯算子的谱特征在无序介质中随机游走尺度极限推导中的作用。第二个项目研究了引入非吸引势对随机梯度场静力学和动力学的影响。第三个项目的重点是开发一种基于晶格安德森哈密顿算子的特征值极端阶统计量的随机漫步在随机潜在景观中定位的详细方法。第四个项目利用新思想来控制统计力学模型中界面的刚性。最后的项目概述了一种基于互换性的新方法，以量化单一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系统的理解，在这些系统中，均匀化理论、谱分析、微分方程以及概率方法（如随机分析、极序统计和无序系统理论等）的分析技术发挥着重要作用。设计了一些项目，以促进对概率论和数学物理感兴趣的博士后和研究生的培训和研究。

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.