Mapping of $(2+1)$-dimensional Kardar-Parisi-Zhang growth onto a driven lattice gas model of dimers

We show that a $(2+1)$-dimensional discrete surface growth model exhibiting Kardar-Parisi-Zhang (KPZ) class scaling can be mapped onto a two-dimensional conserved lattice gas model of directed dimers. The KPZ height anisotropy in the surface model corresponds to a driven diffusive motion of the lattice gas dimers. We confirm by numerical simulations that the scaling exponents of the dimer model are in agreement with those of the $(2+1)$-dimensional KPZ class. This opens up the possibility of analyzing growth models via reaction-diffusion models, which allow much more efficient computer simulations.

Figure 1

(Color online) Mapping of the $(1+1)$-dimensional surface growth onto the 1D ASEP model. Surface attachment (with probability $p$) and detachment (with probability $q$) correspond to anisotropic diffusion of particles (bullets) along the 1D base space.

Figure 2

(Color online) Mapping of the $(2+1)$-dimensional surface growth onto the 2D particle model (bullets). Surface attachment (with probability $p$) and detachment (with probability $q$) correspond to Kawasaki exchanges of particles, or to anisotropic diffusion of dimers in the bisectrix direction of the $x$ and $y$ axes. The crossing points of dashed lines show the base sublattice to be updated. Thick solid (dashed) lines on the surface show the $x$ $\left(y\right)$ cross sections, corresponding to the 1D model (Fig. 1).

Figure 3

(Color online) Logarithmic surface growth in the case of up-down symmetry for different sizes $L=64,128,256,512,1024$ (bottom to top). The dashed line shows the fitting with the form (16). Inset: width saturation values for different system sizes $L$ in the long-time limit.

Figure 4

(Color online) Scaling collapse for $p=1$, $q=0$ with $(2+1)$-dimensional KPZ class exponents for different sizes $L=64,128,256,512,1024$ (bottom to top).