In this extended paper, a clear overview of over 25 years of work which culminated in 2010 in the discovery of the probability distribution for the solution to the Kardar-Parisi-Zhang (KPZ) stochastic PDE is given. The main focus is on
– growth processes and their relationship to the KPZ;
– random growth interfaces and interacting particle systems;
– directed polymers in random media;
– non-linears SPDEs, the stochastic heat equation (SHE) with multiplicative noise and the stochastic Burgers equation.
The author emphasizes three aspects, namely, (1) the approximation of the KPZ equation through a weakly asymmetric simple exclusion process (WASEP); (2) the derivation of the exact one-point distribution of the solution to the KPZ equation with narrow wedge initial data; (3) connections with directed polymers in random media (especially Section 4).
This paper is divided into four sections. In the introduction, the main results and ideas associated with the KPZ equation and its universality class are given. There the reader can find a detailed, comprehensive explanation of the KPZ universality class, e.g., models in this class, the description of polymers. Section 2 gives the rigorous connection between the WASEP and the KPZ equation with a step-by-step explanation how one can obtain this connection. The next section shows how one can derive the exact statistics for the KPZ equation. The final section presents a review of the theory of directed polymers in random media and that the continuum directed random polymer (CDRP) is an universal scaling limit for a wide class of polymer models.