The fixed point of the KPZ universality

KPZ 普适性的不动点

• 批准号: EP/X03237X/1
• 负责人: Nikolaos Zygouras
• 金额: $ 10.27万
• 依托单位: University of Warwick
• 依托单位国家: 英国
• 项目类别: Research Grant
• 财政年份: 2023
• 资助国家: 英国
• 起止时间: 2023 至 无数据
• 项目状态: 未结题
• 来源: https://gtr.ukri.org/projects?ref=EP%2FX03237X%2F1
• 关键词: fixed; point; KPZ; universality

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.

围绕Kardar-Parisi-Zhang（KPZ）普适性的愿景是建立一个随机增长系统的波动理论，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，以取代标准的中心极限理论。驱动力在这方面的努力一直到目前为止的KPZ方程，一个非线性随机偏微分方程，在1986年提出的Kardar，Parisi和张作为普遍的，连续对象在这一类。最近，Matetski-Quastel-Remenik [Acta Mathematica，2022]提出了一个新的观点，即描述这类模型时空统计的中心连续体对象不是KPZ方程，而是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--完全不对称排斥过程（TASEP）的一个合适的标度极限。对这一建设的关键是明确的行列式结构的TASEP，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：可以扩大范围的KPZ不动点，包括在其域的吸引力`非决定性'的过程？KPZ定点的详细统计数据是什么？该项目旨在通过以下方式扩展这一新对象的范围并理解其统计性质：1）在更广泛的一类KPZ模型中建立收敛到KPZ不动点的途径，2）导出KPZ不动点的时间统计。该项目更广泛的前景是理解KPZ不动点与KPZ结构之间的联系。