The one-dimensional KPZ equation and its universality class

TL;DR

This paper reviews recent progress in understanding the one-dimensional KPZ equation, a fundamental stochastic PDE modeling surface growth, focusing on its universality class and related lattice models.

Contribution

It offers a non-technical overview of the advances in the stochastic PDE and lattice models approximating the KPZ equation over the past five years.

Findings

Significant progress in understanding the KPZ universality class

Connection between stochastic PDE and lattice models clarified

Recent analytical and numerical results summarized

Abstract

Our understanding of the one-dimensional KPZ equation, \textit{alias} noisy Burgers equation, has advanced substantially over the past five years. We provide a non-technical review, where we limit ourselves to the stochastic PDE and lattice type models approximating it.