Skewness in $(1+1)$-dimensional Kardar-Parisi-Zhang–type growth

We use the $(1+1)$-dimensional Kardar-Parisi-Zhang equation driven by a Gaussian white noise and employ the dynamic renormalization-group of Yakhot and Orszag without rescaling [J. Sci. Comput. 1, 3 (1986)]. Hence we calculate the second- and third-order moments of height distribution using the diagrammatic method in the large-scale and long-time limits. The moments so calculated lead to the value $S=0.3237$ for the skewness. This value is comparable with numerical and experimental estimates.

Figure 1

Self-energy correction. The self-energy $\Sigma (\mathbf{k},\omega )$ corresponds to the loop. Propagators are indicated by arrowed lines and correlation by a wiggly line.

Figure 2

Perturbation expansion of the correlation $Q(\mathbf{k},\omega )$ to one-loop order.

Figure 3

The third-order moment.