Dimensional crossover in Kardar-Parisi-Zhang growth

TL;DR

This paper investigates the dimensional crossover in 2D KPZ growth models, revealing how surface roughness and height distributions transition from 2D to 1D behavior depending on substrate geometry and size, with implications for solving KPZ models.

Contribution

The study uncovers the mechanisms and scaling laws governing the 2D-to-1D crossover in KPZ growth, including roughness, height distributions, and growth velocity, through extensive simulations.

Findings

Roughness scales as t^{β_{2D}} for t << t_c and t^{β_{1D}} for t >> t_c.

Height distributions transition from 2D flat or cylindrical to Tracy-Widom GOE or GUE.

Universal HDs interpolate between 2D and 1D as L_y/L_x increases.

Abstract

Two-dimensional (2D) KPZ growth is usually investigated on substrates of lateral sizes $L_x=L_y$, so that $L_x$ and the correlation length ($\xi$) are the only relevant lengths determining the scaling behavior. However, in cylindrical geometry, as well as in flat rectangular substrates $L_x \neq L_y$ and, thus, the surfaces can become correlated in a single direction, when $\xi \sim L_x \ll L_y$. From extensive simulations of several KPZ models, we demonstrate that this yields a dimensional crossover in their dynamics, with the roughness scaling as $W \sim t^{\beta_{\text{2D}}}$ for $t \ll t_c$ and $W \sim t^{\beta_{\text{1D}}}$ for $t \gg t_c$, where $t_c \sim L_x^{1/z_{2\text{D}}}$. 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 $L_y/L_x$ increases. Importantly, the crossover scalings are fully determined and indicate a possible way to solve 2D KPZ models.