CAREER: Probability and Mathematical Statistical Mechanics

Probability theory is the mathematical study of random processes. Mathematical statistical mechanics uses probability methods to understand physical phenomena such as transitions from solid to liquid and gaseous phases of matter, or the flow of fluid through a porous medium. This project will focus on two central problems in mathematical statistical mechanics relating to (a) distances in random landscapes and (b) the large scale behavior of a simplified model of a phase transition on high-dimensional lattices. Educational activities will include training of graduate students, organization of a summer school, and the development of probability and statistics learning materials suitable for teaching adults in correctional environments. The first direction of research will be investigation of universal fluctuations in stationary Kardar-Parisi-Zhang models using robust coupling techniques, focusing especially on interacting diffusions and polymer models. The goal is to investigate the extent to which coupling methods apply to general models expected to lie in the KPZ class. A first step will be the study, via analytic methods, of triangular systems relating to multi-path extensions of the O'Connell-Yor semi-discrete polymer, and their scaling at the edge of the limit shape. A second objective is the construction of scaling limits in high dimensional percolation, including joint scaling limits of distances in high dimensional percolation. One goal is to obtain the joint scaling limit of distances to the origin for collections of points at macroscopic distance. This is expected to match the corresponding limit for branching random walk. Another scaling limit of interest is that of the cluster measure. These investigations will be enabled by cluster extension and decoupling techniques akin to those existing for 2-dimensional percolation, but which nonetheless rest on completely different mechanisms in high dimensions. 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.