Universality in two-dimensional Kardar-Parisi-Zhang growth

We analyze simulation results of a model proposed for etching of a crystalline solid and results of other discrete models in the $\mathrm{}(2+1)\mathrm{}$-dimensional Kardar-Parisi-Zhang (KPZ) class. In the steady states, the moments $W_n$ of orders $n=2,3,4\mathrm{}$ of the height distribution are estimated. Results for the etching model, the ballistic deposition model, and the temperature-dependent body-centered restricted solid-on-solid model suggest the universality of the absolute value of the skewness $S\equiv W_3{/W}_2^{3/2}$ and of the value of the kurtosis $Q\equiv W_4{/W}_2^2-3.$ The sign of the skewness is the same as of the parameter $\lambda$ of the KPZ equation which represents the process in the continuum limit. The best numerical estimates, obtained from the etching model, are $|S|=0.26\mathrm{}\pm 0.01$ and $Q=0.134\mathrm{}\pm \mathrm{0.015.}$ For this model, the roughness exponent $\alpha =0.383\mathrm{}\pm 0.008$ is obtained, accounting for a constant correction term (intrinsic width) in the scaling of the squared interface width. This value is slightly below previous estimates of extensive simulations and rules out the proposal of the exact value $\alpha =2/5.\mathrm{}$ The conclusion is supported by results for the ballistic deposition model. Independent estimates of the dynamical exponent and of the growth exponent are $1.605<~z<~1.64$ and $\beta =0.229\mathrm{}\pm 0.005,$ respectively, which are consistent with the relations $\alpha +z=2\mathrm{}$ and $z=\alpha /\beta .$