Gibbs measures in KPZ university class and beyond

University Of Illinois At Urbana-Champaign, Urbana IL

Investigators

Abstract

This project will investigate the Kardar-Parisi-Zhang (KPZ) equation and related stochastic processes from the perspective of Gibbs measures. The KPZ equation, which was introduced by M. Kardar, G. Parisi, and Y.-C. Zhang in 1986, quickly became the default model for random interface growth in physics. The KPZ equation (and its extension, the KPZ line ensemble) inherits a remarkable structure called the Gibbs property that represents an inner consistency. The main goal of the project is to leverage the Gibbs property to study the KPZ equation, with a focus on constructing the solution and understanding long-range correlation. Students will participate in the research and the awardee plans to co-organize workshops and seminars and to write survey articles. This project involves three related directions of research. The first investigates the relation between the KPZ equation and the KPZ line ensemble, and plans to establish connections based on Gibbs properties.. The second research direction is to establish the universality of the Airy line ensemble. The Airy line ensemble has been shown to converge for various models such as the PNG droplet, Dyson's Brownian motion, exponential/geometric last passage percolation, and lozenge tilings, primarily due to their integrable nature. This project seeks to demonstrate the universality of the Airy line ensemble under broader and milder assumptions. The third direction is to use Gibbsian line ensembles to study the Laguerre unitary ensemble (LUE), which lies outside the KPZ universality class, building on the successful construction of the Bessel line ensemble and exploring its many potential applications. 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.