Disordered Systems and Quasi-Invariant Dynamics

统计物理学提供了一套非常有用的工具来分析涉及复杂相互作用系统的自然现象，这些系统的动力学很难准确描述，但其平均行为可以进行分析。它的应用范围远远超出了物理学领域。为了更深入地理解这一理论的数学基础，有必要研究简化的模型，这些模型仍然表现出与物理系统相似的丰富行为。本研究计画将研究其中一类简化离散模型--逾渗模型。在另一个独立的主题领域，受统计物理学启发的概率工具将应用于微分方程，以分析其解的长期行为。该项目将研究具有物理意义的度量的行为，包括临界随机环境中的化学距离和有效电阻，例如Bernoulli渗透和低维定向渗透。例如，这些度量对于随机游走的研究至关重要。另外，该项目还将研究具有随机初始数据的非线性Hamilton偏微分和随机微分方程的行为，特别是从Gibbs类型的不变和准不变测度中选择的数据。这些对象允许构造特殊解决方案，与一般情况相比，具有改进的长期行为。首席研究员将参与暑期学校的组织，以及设计和教授关于晶格系统（如渗流）和连续动力系统（如随机数据的偏微分方程）的统计力学课程。该奖项反映了NSF的法定使命，并通过使用基金会的智力价值和更广泛的影响审查标准进行评估，被认为值得支持。

Statistical physics provides an immensely useful set of tools to analyze natural phenomena involving complex interacting systems whose dynamics is difficult to describe exactly, but whose average behavior is amenable to analysis. It has found applications far beyond the realm of physics. To gain a deeper understanding of the mathematical underpinnings of this theory, it is necessary to study simplified models that nevertheless exhibit rich behavior similar to physical systems. This research project will study one class of such simplified discrete models, percolation models. In a separate topic area, probabilistic tools inspired by statistical physics will be applied to differential equations, to analyze the long-time behavior of their solutions.The project will study the behavior of metrics of physical significance, including the chemical distance and the effective resistance in critical random environments, such as Bernoulli percolation and oriented percolation in low dimensions. These metrics are of central importance for the study of random walks, for instance. Separately, the project will also study the behavior of nonlinear Hamiltonian partial differential and stochastic differential equations with random initial data, especially data chosen from invariant and quasi-invariant measures of Gibbs type. These objects allow for the construction of special solutions with improved long-time behavior compared to the generic case. The principal investigator will participate in the organization of summer schools, as well as the design and teaching of classes on statistical mechanics of both lattice systems like percolation, and continuous dynamical systems like partial differential equations with random data.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.