Upper Critical Dimension, Dynamic Exponent, and Scaling Functions in the Mode-Coupling Theory for the Kardar-Parisi-Zhang Equation
We study the mode-coupling approximation for the Kardar-Parisi-Zhang equation in the strong-coupling regime. By constructing an ansatz consistent with the asymptotic forms of the correlation and response functions we determine the upper critical dimension dc = 4 and the expansion z = 2-\(d-4\)/4+O\(\(4-d\)2\) around dc. We find the exact z = 3/2 value in d = 1, and estimate the values z~=1.62, z~=1.78 in d = 2, 3. The result dc = 4 and the expansion around dc are very robust and can be derived just from a mild assumption on the relative scale on which the response and correlation functions vary as z approaches 2.