Scaling Limits of Interacting Particle Systems and the KPZ Universality Class in Non-Equilibrium Dynamics

This monograph provides a rigorous analysis of the scaling limits of one-dimensional interacting particle systems, specifically focusing on the Asymmetric Simple Exclusion Process (ASEP) and its convergence to the Kardar-Parisi-Zhang (KPZ) universality class. We examine the mathematical derivation of the stochastic partial differential equation (SPDE) governing interface growth, deriving the KPZ equation via the weakly asymmetric limit of the microscopic height function. Special attention is given to the ill-posed nature of the non-linear term (x h)2 and the necessity of renormalization, as treated in the theory of regularity structures. Furthermore, we discuss the integrability of the model through the Cole-Hopf transformation, mapping the KPZ equation to the Stochastic Heat Equation (SHE). We conclude by establishing the connection between the long-time fluctuations of the KPZ height field and the Tracy-Widom distributions arising from Random Matrix Theory, thereby characterizing the universal scaling exponents 1:2:3 inherent to the KPZ class.