Absence of Finite Upper Critical Dimension in the Spherical Kardar-Parisi-Zhang Model

It is shown in this paper that the scaling properties of an $N$-component Kardar-Parisi-Zhang equation in the large $N$ limit (spherical limit) can be obtained by solving the mode-coupling equation. The full scaling functions and the critical exponents in the large $N$ limit are obtained exactly by solving the mode-coupling equation numerically. The dynamic exponent $z$ is found to be 1.615 at dimension $d=2\mathrm{}$, very close to the value obtained by numerical simulation of growth process, but $z$ remains less than 2 in any finite dimension, suggesting the existence of a strong coupling regime for any finite dimension.