On the Renormalization of the Kardar-Parisi-Zhang equation

TL;DR

This paper applies renormalization techniques to the KPZ equation, revealing critical dimensions and exact exponents, and discusses the behavior of correlation functions in the strong-coupling phase.

Contribution

It provides an exact renormalization group analysis of the KPZ equation near its upper critical dimension, identifying critical exponents and singularities.

Findings

Renormalization group yields exact exponents for d > 2

Identifies d=4 as the upper critical dimension

Correlation functions develop an essential singularity at strong coupling

Abstract

The Kardar-Parisi-Zhang (KPZ) equation of nonlinear stochastic growth in d dimensions is studied using the mapping onto a system of directed polymers in a quenched random medium. The polymer problem is renormalized exactly in a minimally subtracted perturbation expansion about d = 2. For the KPZ roughening transition in dimensions d > 2, this renormalization group yields the dynamic exponent z* = 2 and the roughness exponent chi* = 0, which are exact to all orders in epsilon = (2 - d)/2. The expansion becomes singular in d = 4, which is hence identified with the upper critical dimension of the KPZ equation. The implications of this perturbation theory for the strong-coupling phase are discussed. In particular, it is shown that the correlation functions and the coupling constant defined in minimal subtraction develop an essential singularity at the strong-coupling fixed point.