Noncommutative statistical mechanics: probability at the confluence

非对称简单排除过程(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ASEP型过程与著名的正交多项式族之间正在出现的平行关系，暗示了一种隐藏的概率现象，有望极大地扩展ASEP型过程的适用性。关键在于一种常见的计算伪像，称为矩阵ansatz。在这个项目中，我们基于它的矩阵ANSATZ和非对易无限维现象之间的关系，探索了关于ASEP的一个新的视角。通过这种方式，我们的目标是改变经典概率理论和非对易概率理论之间的界面，对ASEP达成更基本的理解，并为一个长期存在的数学分析的开放问题--自由群因子同构问题开辟创新的途径。

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.