Studies in Probability Theory and Statistical Physics

University Of Maryland, College Park, College Park MD

Investigators

Abstract

The project focuses on the behavior of disordered systems in statistical physics, which arise when studying physical systems in inhomogeneous media. The research aims to understand the effects of inhomogeneity on the systems’ behavior. Focus is given to membranes in random environments and the study of their fluctuations. Such membranes arise naturally when classical models, such as the Ising or XY models, are either assigned random inhomogeneous interactions or are placed in a random magnetic field. They have also recently been discussed in the study of random quantum circuits. The research aims to explore the emergent physical phenomena from a mathematically rigorous standpoint. The project also provides research opportunities for graduate students. The large-scale properties of equilibrium statistical physics systems, such as the classical Ising or XY models, can change significantly when the models are placed in an inhomogeneous environment. Such environments are often modeled by quenched (frozen-in) disorder and lead to random-bond or random-field versions of the classical systems. Another instance of this phenomenon is first-passage percolation, where the metric structure of the space is altered by random local perturbations. This project will study fundamental problems in these areas with rigorous mathematical tools. The problems are centered on minimal surfaces in random environments, which are models for both the interfaces between different spins in random-bond and random-field Ising models, for the spins of the random-field XY model, and for geodesics in first-passage percolation. The project aims to capitalize on recent progress to obtain an in-depth analysis of the properties of minimal surfaces in random environments. 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.