Probability distributions for kinetic roughening in the Kardar-Parisi-Zhang growth with long-range temporal and spatial correlations

TL;DR

This study numerically explores how long-range temporal and spatial correlations affect the probability distributions of interface width and height in the KPZ growth model, revealing distinct impacts on distribution forms and transitions.

Contribution

It provides new insights into the effects of long-range correlations on distribution forms in KPZ growth, including the transition from Tracy-Widom to Gaussian distributions for height.

Findings

Long-range temporal correlations do not significantly alter the width distribution, which remains approximately lognormal.

Spatial correlations influence the width distribution, making it more asymmetric and fat-tailed beyond a certain correlation threshold.

Height distributions transition from Tracy-Widom to Gaussian forms depending on the type and strength of correlations.

Abstract

We investigate numerically the effects of long-range temporal and spatial correlations based on the rescaled distributions of the squared interface width $W^2(L,t)$ and the interface height $h(x,t)$ in the (1+1)-dimensional Kardar-Parisi-Zhang (KPZ) growth system within the early growth regimes. Through extensive numerical simulations, we find that long-range temporally correlated noise could not significantly impact the distribution form of the interface width. Generally, $W^2(L,t)$ obeys approximately lognormal distribution when the temporal correlation exponent $\theta \ge 0$. On the other hand, the effects of long-range spatially correlated noise are evidently different from the temporally correlated case. Our results show that, when the spatial correlation exponent $\rho \le 0.20$, the distribution forms of $W^2(L,t)$ approach the lognormal distribution, and when $\rho > 0.20$, the distribution becomes more asymmetric, steep, and fat-tailed, and tends to an unknown distribution form. As a comparison, probability distributions of the interface height are also provided in the temporally and spatially correlated KPZ system, exhibiting quite different characteristics from each other within the whole correlated regimes. For the temporal correlation, the height distributions satisfy Tracy-Widom Gaussian orthogonal ensemble (TW-GOE) when $\theta \to 0$, and with increasing $\theta$, the height distributions crossover continously to an unknown distribution. However, for the spatial correlation, the height distributions gradually transition from the TW-GOE distribution to the standard Gaussian form.