Integrable probability and random matrices: 2d structures, limit theorems

本研究的目的是为了更好地理解一类与统计力学、随机矩阵理论和表示理论相关的随机系统的极限行为。这类系统的例子包括三维空间中的随机阶梯表面，特别是模拟融化的晶体；方形冰模型，这是对薄冰的数学二维近似；以及用于模拟的相互作用的粒子系统(例如，单车道公路、海洋中浮游生物的生长)。我们的目标是从超大型系统的微观定义出发，提取它们的宏观性质，重点放在普遍随机场和分布的出现上。我们研究的系统有两个主要区别。首先，我们集中研究和探索了二维结构，这些结构推广了许多经典的一维概率模型，例如随机矩阵的特征值。我们所考虑的二维延拓给出了早期一维结果的新的且往往更自然的解释，也为证明关于著名的一维模型的新的有趣的渐近结果铺平了道路。其次，本研究中的大多数模型都具有丰富的代数结构，这通常意味着可以以简明的方式计算许多可观测对象的预期。这些技术包括使用表示论起源的对称函数、差分算子的特征函数、正交多项式等。这种精确的可解性为精细的渐近分析提供了工具，并提供了出现在极限中的普遍物体的性质，如Tracy-Widom分布、特征值分布、格角过程(其二维扩展)和高斯自由场。对于精确可解模型所得到的结果通常被认为可以推广到许多相似的随机系统。

The goal of this research is to achieve a better understanding of the limit behavior of a class of stochastic systems related to statistical mechanics, random matrix theory and representation theory. Examples of such systems include random stepped surfaces in three-dimensional space which, in particular, model melting crystals; square-ice model which is a mathematical two-dimensional approximation for a thin layer of ice; and interacting particle systems used for modeling (e.g., a one-lane highway, the growth of plankton in the ocean). Our aim is to extract macroscopic properties of very large systems starting from their microscopic definitions, with main accents on the appearance of the universal random fields and distributions.There are two main distinctions of the systems we study. First is that we concentrate on and explore 2d structures which generalize many classical 1d probabilistic models such as eigenvalues of a random matrix. The two-dimensional extensions that we consider give new and often more natural interpretations of earlier one-dimensional results and also pave the way to prove new interesting asymptotic results about well-known one-dimensional models. Second, most models in this research enjoy a rich algebraic structure, which usually means that expectations of many observables can be computed in a concise manner. The techniques include usage of symmetric functions of representation-theoretic origin, eigenfunctions of difference operators, orthogonal polynomials, etc. This exact solvability provides tools for delicate asymptotic analysis and gives access to the properties of the universal objects appearing in the limit, such as Tracy-Widom distributions, the GUE-eigenvalues distribution, the GUE-corners process (its two-dimensional extension), and the Gaussian Free Field. The results obtained for the exactly solvable models are generally believed to extend to a large variety of similar stochastic systems.