Existence of the upper critical dimension of the Kardar–Parisi–Zhang equation

Abstract

The controversy whether or not the Kardar–Parisi–Zhang (KPZ) equation has an upper critical dimension (UCD) is going on for quite a long time. Some approximate integral equations for the two-point function served as an indication for the existence of a UCD, by obtaining a dimension, above which the equation does not have a strong coupling solution. A surprising aspect of these studies, however, is that various authors who considered the same equation produced large variations in the UCD. This caused some doubts concerning the existence of a UCD. Here we revisit these calculations, describe the reason for such large variations in the results of identical calculations, show by a large- d asymptotic expansion that indeed there exists a UCD and then obtain it numerically by properly defining the integrals involved. Since many difficult problems in condensed matter physics of non-linear nature are handled with mode-coupling and self-consistent theories, this work might also contribute to other researchers working on a large class of different problems that might run into the same inconsistencies.