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

本研究的目的是更好地理解与统计力学、随机矩阵理论和表示论有关的一类随机系统的极限行为。这种系统的示例包括三维空间中的随机阶梯表面，其特别地对熔化的晶体进行建模;方形冰模型，其是薄冰层的数学二维近似;以及用于建模的相互作用粒子系统（例如，一条单车道的高速公路，海洋中浮游生物的生长）。我们的目标是从微观定义出发，提取大系统的宏观性质，主要强调泛随机场和分布的出现。我们研究的系统有两个主要区别。首先，我们专注于和探索二维结构，推广了许多经典的一维概率模型，如随机矩阵的特征值。我们考虑的二维扩展为早期的一维结果提供了新的且通常更自然的解释，并且还为证明有关已知一维模型的新的有趣的渐进结果铺平了道路。其次，本研究中的大多数模型都具有丰富的代数结构，这通常意味着许多可观测量的期望值可以以简洁的方式计算。这种精确可解性为精细的渐近分析提供了工具，并提供了对出现在极限中的普适对象的性质的访问，例如Tracy-Widom分布，GUE特征值分布，GUE角点过程（它的二维扩展），和高斯自由场。对于精确可解的模型所得到的结果一般被认为可以推广到许多类似的随机系统。

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.