Asymptotics of Positive Temperature Models From Statistical Mechanics

This project aims to understand the behavior of large random systems at positive temperature. Examples of such models are given by random three-dimensional stepped surfaces, interacting particles systems, the six-vertex model (used to describe the structure of a thin film of water molecules), and interacting avoiding random walkers (sometimes called line ensembles).

Many of these models have remarkable algebraic and combinatorial properties, which makes their study tractable. The main goal of the project is to obtain a detailed description (involving exact mathematical formulas) of the asymptotic behavior of these random systems as their size (volume and/or the number of particles) grows.

The project involves three interwined directions of research. The first involves the derivation and asymptotic analysis of joint observables for integrable models in the Kardar-Parisi-Zhang (KPZ) universality class. A distinguished feature of integrable models is that they allow for various exact formulas of different observables.

A notorious problem with many of these formulas is that they are difficult to study asymptotically, due to the presence of hard to control cross-terms, and one of the goals of the project is to develop a framework for analyzing these cross-terms. The second direction of the project is to establish universal scaling limits for Gibbsian line ensembles.

Various integrable models, such as Hall-Littlewood processes and the log-gamma polymer, naturally carry a structure of a line ensemble and it is expected that these ensembles converge to the parabolic Airy line ensemble (one of the universal scaling limits in the KPZ universality class) -- the project seeks to establish this statement. The third direction of the project is to utilize loop equations to understand the scaling limits of multi-level interacting particle systems, which are discrete analogues of the beta-corners processes from random matrix theory and are related to Jack symmetric functions.

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.