Disordered Systems and Quasi-Invariant Dynamics

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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.