Asymptotics of Positive Temperature Models From Statistical Mechanics

该项目旨在了解大型随机系统在正温度下的行为。这类模型的例子有随机三维阶梯表面、相互作用的粒子系统、六顶点模型(用于描述水分子薄膜的结构)和相互作用的避免随机行走的人(有时称为线系综)。这些模型中有许多具有显著的代数和组合性质，这使得它们的研究变得容易进行。该项目的主要目标是获得这些随机系统随着其大小(体积和/或粒子数量)增长而渐近行为的详细描述(包括精确的数学公式)。该项目涉及三个相互交织的研究方向。第一部分是关于Kardar-Parisi-Zhang(KPZ)普适性类中可积模型联合观测的推导和渐近分析。可积模型的一个显著特点是，它们允许不同的观察量的不同的精确公式。其中许多公式的一个臭名昭著的问题是，由于交叉项的存在很难控制，它们很难渐近研究，该项目的目标之一是开发一个分析这些交叉项的框架。该项目的第二个方向是为吉布斯线谱建立通用的比例限制。各种可积模型，如Hall-Littlewood过程和对数-伽马聚合物，自然具有线系综结构，预计这些系综会收敛到抛物线艾利线系综(KPZ普适性类中的普遍标度限制之一)--该项目试图建立这一说法。该项目的第三个方向是利用循环方程来了解多级相互作用粒子系统的比例极限，这是随机矩阵理论中的Beta-Corner过程的离散类似物，与Jack对称函数有关。该奖项反映了NSF的法定使命，并通过使用基金会的智力优势和更广泛的影响审查标准进行评估，被认为值得支持。

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.