Dimensional crossover in Kardar-Parisi-Zhang growth

Two-dimensional (2D) KPZ growth is usually investigated on substrates of lateral sizes Lₓ=Ly, so that Lₓ and the correlation length (ξ) are the only relevant lengths determining the scaling behavior. However, in cylindrical geometry, as well as in flat rectangular substrates Lₓ ≠ Ly and, thus, the surfaces can become correlated in a single direction, when ξ ~ Lₓ Ly. From extensive simulations of several KPZ models, we demonstrate that this yields a dimensional crossover in their dynamics, with the roughness scaling as W ~ t^β2D for t tc and W ~ t^β1D for t tc, where tc ~ Lₓ^1/z2D. The height distributions (HDs) also cross over from the 2D flat [cylindrical] HD to the asymptotic Tracy-Widom GOE [GUE] distribution. Moreover, 2D-to-1D crossovers are found also in the asymptotic growth velocity and in the steady state regime of flat systems, where a family of universal HDs exists, interpolating between the 2D and 1D ones as Ly/Lₓ increases. Importantly, the crossover scalings are fully determined and indicate a possible way to solve 2D KPZ models.