The exact results presented in the previous chapter allow us to obtain the scaling exponents for d = 1, and reduce the number of independent scaling exponents to one. The same results can be obtained using the dynamic renormalization group method, which we now develop and use to study the scaling properties of the KPZ equation. In particular, we analyze the ‘flow equations’ and extract the exponents describing the KPZ universality class for d = 1. We also discuss numerical results leading to the values of the scaling exponents for higher dimensions.

Basic concepts

So far, we have argued that we can distinguish between various growth models based on the values of the scaling exponents α, β and z. The existence of universal scaling exponents and their calculation for various systems is a central problem of statistical mechanics. A main goal for many years has been to calculate the exponents for the Ising model, a simple spin model that captures the essential features of many magnetic systems. A major breakthrough occurred in 1971, when Wilson introduced the renormalization group (RG) method to permit a systematic calculation of the scaling exponents. Since then the RG has been applied successfully to a large number of interacting systems, by now becoming one of the standard tools of statistical mechanics and condensed matter physics. Depending on the mathematical technique used to obtain the scaling exponents, the RG methods can be partitioned into two main classes: real space and k-space (Fourier space) RG.