Appears in collection : 2025 IHES Summer School – Statistical Aspects of Nonlinear Physics

KPZ scaling leads to a natural renormalization of the entire space-time trajectory of random interface growth models. What is the universal limit of this process? I will first introduce the KPZ fixed point and directed landscape which characterize this limit. Then, I will describe some implications of this full scaling limit, in particular to the theory of fluctuating hydrodynamics and multispecies interacting particle systems. Finally, I will sketch how the Yang-Baxter equation -- the fundamental relation from quantum integrable systems -- gives the key to unlocking this scaling limit.