The fixed point of the KPZ universality

围绕Kardar-Parisi-Zhang(KPZ)普适性的愿景是为随机增长的系统建立一个涨落理论，从而通过理解随机性和可积性之间的相互作用，为强相关系统创建一个框架，取代标准的中心极限理论。到目前为止，这一努力的推动力是KPZ方程，这是一种非线性随机偏微分方程，由Kardar，Parisi和Zhang于1986年提出，作为这类普遍的、连续的对象。最近，Matetski-Quastel-Remenik[数学学报，2022]提出，捕捉这类模型全部时空统计的中心连续介质对象不是KPZ方程，而是所谓的KPZ不动点，KPZ不动点被构造为相互作用粒子系统的一个中心模型的合适的尺度极限，即完全不对称排斥过程(TASEP)。这种构造的关键是TASEP的显式行列式结构，它允许使用强大的行列式演算。现在提出了一些重要的问题：能否通过将“非确定性”过程包括在其吸引域中来扩大KPZ不动点的范围？关于KPZ固定点的详细统计数据有哪些？这个项目的目的是扩大这个新对象的范围并了解它的统计性质：1)在更广泛的一类KPZ模型中打开收敛到KPZ不动点的途径；2)推导KPZ不动点的时间统计量。

The vision around the Kardar-Parisi-Zhang (KPZ) universality is to build a theory of fluctuations for randomly growing systems and thus to create a framework, in place of the standard central limit theory, for strongly correlated systems via understanding the interplay between randomness and integrability. The driving force in this endeavour has been so far the KPZ equation, a nonlinear stochastic partial differential equation, proposed in 1986 by Kardar, Parisi and Zhang as the universal, continuum object in this class. Recently, it has been proposed by Matetski-Quastel-Remenik [Acta Mathematica, 2022] that the central, continuum object that captures the full space-time statistics of models in the class is not the KPZ equation but rather the so called KPZ Fixed Point.The KPZ Fixed Point was constructed as a suitable scaling limit of one central model of interacting particle systems, the Totally Asymmetric Exclusion Process (TASEP). Crucial towards this construction has been the explicit determinantal structure of TASEP, which allows for the use of powerful determinantal calculus. Some important questions are now raised: Can one broaden the scope of the KPZ fixed point by including in its domain of attraction `non-determinantal' processes? What are the detailed statistics of the KPZ fixed point? This project aims at expanding the scope of this new object and at understanding its statistical properties by:1) opening the route to establishing convergence to the KPZ fixed point within a broader class of KPZ models,2) deriving the temporal statistics of the KPZ fixed point.The broader outlook of the project is the understanding of the connections between the KPZ fixed point and KPZ structures to integrability.