Universality Classes of Integrable Lattice Models in the Scaling Limit of the KPZ Fixed Point

The Kardar-Parisi-Zhang (KPZ) universality class encompasses a broad spectrum of non-equilibrium stochastic growth models in (1+1)-dimensions, characterized by the dynamical scaling exponent z=3/2 and the roughness exponent =1/2. This monograph rigorously examines the convergence of integrable lattice models, specifically the Totally Asymmetric Simple Exclusion Process (TASEP) and the Stochastic Six-Vertex (S6V) model, to the KPZ fixed point—a scale-invariant Markov process constructed by Matetski, Quastel, and Remenik (2021). We analyze the exact Fredholm determinant representations of the transition probabilities for these lattice models and demonstrate their asymptotic reduction to the universal kernel of the KPZ fixed point under the critical 1:2:3 scaling. Furthermore, we discuss the algebraic structure of Macdonald processes as a unifying framework for integrability, establishing the universality of the Tracy-Widom statistics and the Airy sheet in the scaling limit. The results confirm the strong KPZ universality conjecture for this class of integrable probabilistic systems.