Noncommutative statistical mechanics: probability at the confluence

非对称简单排斥过程（英语：asymmetric simple exclusion process，ASEP）是统计力学中的一个基本数学结构，它结合了随机运动的概念和物质不能在同一位置共存的物理要求。因此，它具有广泛的应用，从交通和流体流动到生物过程。然而，最近的进展，特别是新兴的并行性之间的ASEP型过程和著名的正交多项式的家庭，暗示在一个隐藏的概率现象，有望大大扩展的ASEP的适用性。关键是在一个常见的计算工件，称为矩阵analog。在这个项目中，我们探索了一个新的角度对ASEP，基于其矩阵矩阵和非交换无穷维现象之间的关系。以这种方式，我们的目标是转换之间的接口的“经典”和非交换概率论，到达一个更根本的理解ASEP，并开辟了创新的方法，一个长期存在的开放问题的数学分析，自由群因子同构问题。

The asymmetric simple exclusion process (ASEP) is a fundamental mathematical construction in statistical mechanics that combines the notion of stochastic motion with a physical requirement that matter cannot coexist in the same location. As such, it has wide-ranging applications, from traffic and fluid flow to biological processes. However, recent advances, particularly the emerging parallelism between ASEP-type processes and well-known families of orthogonal polynomials, hint at a hidden probabilistic phenomenon that promises to greatly expand the applicability of the ASEP. The key is in a common computational artifact, referred to as the matrix ansatz. In this project, we explore a new perspective on the ASEP, based on the relationship between its matrix ansatz and noncommutative infinite-dimensional phenomena. In this manner, we aim to transform the interface between the 'classical' and noncommutative probability theories, arrive at a more fundamental understanding of the ASEP, and open up innovative approaches to a long-standing open question of mathematical analysis, the free group factors isomorphism problem.