Universality Classes for Directed Polymers in Random Media and the KPZ Fixed Point

This monograph explores the rigorous mathematical foundations of the Kardar-Parisi-Zhang (KPZ) universality class, with a specific focus on Directed Polymers in Random Media (DPRM) in 1+1 dimensions. We analyze the asymptotic behavior of the free energy fluctuations of the polymer model, demonstrating their convergence to the Tracy-Widom distributions of Random Matrix Theory under the characteristic t1/3 scaling. Furthermore, we discuss the recent construction of the KPZ fixed point as a scaling invariant Markov process, as established by Matetski, Quastel, and Remenik. By utilizing the framework of integrable probability and the geometric RSK correspondence, we elucidate the connection between the discrete polymer partition functions and the continuum Airy sheet, providing a unified description of the universal scaling limits governing interface growth and stochastic transport.