On Growth, Disorder, and Field Theory

TL;DR

This paper reviews recent advances in statistical field theory for non-equilibrium systems, focusing on the KPZ equation, stochastic Burgers equation, and directed polymers, highlighting nonperturbative scale-invariance and anomalous scaling behaviors.

Contribution

It introduces a self-consistent asymptotic theory based on operator product expansion to understand strong disorder regimes in non-equilibrium growth models.

Findings

Identification of nonperturbative scale-invariance at strong disorder

Proposal of a self-consistent asymptotic theory for scaling exponents

Discovery of dynamical anomalies due to dangerous irrelevant couplings

Abstract

This article reviews recent developments in statistical field theory far from equilibrium. It focuses on the Kardar-Parisi-Zhang equation of stochastic surface growth and its mathematical relatives, namely the stochastic Burgers equation in fluid mechanics and directed polymers in a medium with quenched disorder. At strong stochastic driving -- or at strong disorder, respectively -- these systems develop nonperturbative scale-invariance. Presumably exact values of the scaling exponents follow from a self-consistent asymptotic theory. This theory is based on the concept of an operator product expansion formed by the local scaling fields. The key difference to standard Lagrangian field theory is the appearance of a dangerous irrelevant coupling constant generating dynamical anomalies in the continuum limit.