Two-Dimensional KPZ Evolution, Fluctuation Lower Bounds, and Ultrametricity

Stanford University, Stanford CA

Investigators

Abstract

This project concerns several problems in probability theory. One class of problems addresses the evolution of random surfaces according to the KPZ equation, named after its discoverers, Kardar, Parisi, and Zhang. Random surfaces have attracted a lot of recent attention in probability theory, and there are many unanswered questions. This project will aim to answer some of these questions. A second class of problems involves extending and developing a theory of lower bounds on fluctuations of random variables. Understanding fluctuations of random variables is one of the basic goals of probability theory, but there are many important problems where existing methods do not give desirable results. The PI aims to make some progress in this area by providing a new set of tools. Finally, a third class of problems centers around understanding ultrametric spaces that arise in the study of models from statistical mechanics. The strategy of working on problems in varied areas of probability theory at the same time has the potential of uncovering new connections. The problems concerning the KPZ evolution are mainly about producing a solution of the equation in 2D. This would be an important breakthrough because the task of constructing any solution for the 2D KPZ equation has remained intractable so far. The results about fluctuation lower bounds would give the optimal conditions under which the current best lower bounds can be proved for planar growth models. Previously, such lower bounds required restrictive conditions. The proposed method of solution is based on a novel coupling, which may be of independent interest. The research on ultrametricity will shed light on the hierarchical organization of states in spin glass models, especially for models with full replica symmetry breaking. It will also introduce a novel connection between the study of these models and tools from graph theory such as Szemeredi's regularity lemma. 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.