Random Motion in Disordered Media: Surface Growth, Ballisticity, and Trapping

University Of California-Berkeley, Berkeley CA

Investigators

Abstract

This project develops the rigorous theory of statistical mechanics, the study of mathematical models, often random in nature and involving many tiny particles or other bodies, that idealize and exemplify physically interesting phenomena such as the transitions in microscopic structure undergone by a cube of ice as it melts into water. The notion of universality plays a fundamental role: certain mathematical structures emerge from many different random systems when these systems are viewed on an appropriately large scale in space and time. We will develop the theory of randomly growing interfaces, where a one-dimensional surface is growing in a local fashion at random rates while also being subject to restoring forces such as surface tension. In this context, the right notion of universality is specified by a stochastic partial differential equation, the Kardar-Parisi-Zhang (KPZ) equation. Its late-time and large-scale behavior models universal aspects of many randomly growing interfaces. The construction of line ensembles having a natural Gibbs resampling property whose lowest indexed curve is the fundamental solution of the KPZ equation is a technique developed in collaboration with Ivan Corwin that marries integrable systems approaches to probabilistic ideas. The proposal plans to exploit this new perspective to investigate several phenomena in models in the KPZ universality class, including aging and decorrelation in Brownian last passage percolation, and polymer coalescence and tree structure in the Airy sheet.