Stationary State Skewness in Two Dimensional KPZ Type Growth

Abstract

We present numerical Monte Carlo results for the stationary state properties of KPZ type growth in two dimensional surfaces, by evaluating the finite size scaling (FSS) behaviour of the 2nd and 4th moments, $W_2$ and $W_4$, and the skewness, $W_3$, in the Kim-Kosterlitz (KK) and BCSOS model. Our results agree with the stationary state proposed by L\"assig. The roughness exponents $W_n\sim L^{\alpha_n}$ obey power counting, $\alpha_n= n \alpha$, and the amplitude ratio's of the moments are universal. They have the same values in both models: $W_3/W_2^{1.5}= -0.27(1)$ and $W_4/W_2^{2}= +3.15(2)$. Unlike in one dimension, the stationary state skewness is not tunable, but a universal property of the stationary state distribution. The FSS corrections to scaling in the KK model are weak and $\alpha$ converges well to the Kim-Kosterlitz-L\"assig value $\alpha={2/5} $. The FSS corrections to scaling in the BCSOS model are strong. Naive extrapolations yield an smaller value, $\alpha\simeq 0.38(1)$, but are still consistent with $\alpha={2/5}$ if the leading irrelevant corrections to FSS scaling exponent is of order $y_{ir}\simeq -0.6(2)$.