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

We study the mode-coupling theory for the Kardar-Parisi-Zhang equation in the strong-coupling regime. focusing on the long-term 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 k in dimension d is found to behave asymptotically at time t as $C(k,t)\simeqA/k^{d+4-2z} (Btk^z)^\gamma/z exp(-(Btk^z)^{1/z})$ , with $\gamma=(d-1)/2$, A a determined constant, and B a scale factor.