The variational principle, or the least action principle, offers a framework for the study of the Large Deviation Principle (LDP) for a stochastic system. The KPZ equation is a stochastic PDE that is central to a class of random growth phenomena. In this talk, we will study the Freidlin-Wentzell LDP for the KPZ equation through the lens of the variational principle. Such an approach goes under the name of the weak noise theory in physics. We will explain how to extract various limits of the most probable shape of the KPZ equation in the setting of the Freidlin-Wentzell LDP. We will also review the recently discovered connection of the weak noise theory to integrable PDEs.

This talk is based in part on joint works with Pierre Yves Gaudreau Lamarre and Yier Lin.