The fixed point of the KPZ universality

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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.