Stretched exponential relaxation in the mode-coupling theory for the Kardar-Parisi-Zhang equation

We study the mode-coupling theory for the Kardar-Parisi-Zhang equation in the strong-coupling regime, focusing on the long time properties. By a saddle point analysis of the mode-coupling equations, we derive exact results for the correlation function in the long-time limit—a limit that is hard to study using simulations. The correlation function at wave vector $\mathbf{k}$ in dimension d is found to behave asymptotically at time t as $C(\mathbf{k},t)\simeq {A/k}^{d+4\mathrm{}-2z}{(Btk}^z{)}^{\gamma /z}\exp \left[-{(Btk}^z{)}^{1/z}\right],$ with $\gamma =\left(d-1\right)/2,\hspace{0.5em}A$ a determined constant, and B a scale factor.