Universal Coarsening and Giant-Cluster Formation in Growing Interfaces

Abstract

Clusters formed by fluctuations of two-dimensional (2D) directed interfaces around a threshold level have been extensively studied at equilibrium and in nonequilibrium steady states, but their coarsening dynamics remain poorly understood. Here, we numerically investigate this unexplored coarsening of clusters in 2D growing interfaces believed to belong to the Kardar-Parisi-Zhang universality class. Using a two-point spatial correlator, we demonstrate statistical time invariance of the evolving configurations and identify scaling forms shared across distinct models. We reveal a pronounced asymmetry in the growth of the largest clusters: one cluster emerges as a giant structure whose characteristic length exceeds the correlation length. Population-dependent scaling forms for the number densities of cluster areas are uncovered. These findings highlight new universal aspects of growing interfaces and suggest avenues for experimental verification.

Figures (4)

• Figure 1: Instantaneous configurations of the signs of the height fluctuations about their mean, during the kinetic roughening of a growing interface simulated by the SHE-Euler rule at $t=$ (a) 1, (b) 5, (c) 60, (d) 480 (core of the growth regime), (e) 3840 (transition to the steady state), (f) 15360 (steady state). Positive (negative) signs are painted in violet (green) color. The largest cluster of positive (negative) sign is highlighted in magenta (blue) color. Periodic boundaries are employed. The lateral size is $L = 512$.
• Figure 2: Space-time correlations in the SHE-Euler model. (a) Data at different times given in the key, increasing from left to right. Inset: the correlation length versus time. The dashed line is a guide-to-the-eye with slope $1/z$ and $z \approx 1.61$. White and gray areas distinguish the growth regime (GR) and the steady state (SS), respectively. (b) Evidence for dynamic scaling and universality in the GR (blue data above) and SS (green data below). Lines refer to data from (a). Datapoints for the RSOS with $t_{\textrm{lab}} = 1600.5$ (GR), 51200.5 (SS); and ETC with $t_{\textrm{lab}} = 320.5$ (GR), $20480.5$ (SS). Uncertainties correspond to one standard deviation.
• Figure 3: Ensemble-averaged area of the largest cluster of positive (a) and negative (b) sign in the SHE-Euler model. Inset of (a): $A_{l-(+)}$ area of the largest cluster of negative (positive) sign in the RSOS (ETC) model for $L = 512$. Solid lines indicate the asymptotic area fraction, common to all models. Dashed lines are guides-to-the-eye with the slopes depicted. Insertion in (b): same data of the main panel, scaled so as to collapse data at long times. Black dots are guides-to-the-eye that distinguish among segments of the curve. Dotted lines cover the interval of the representative uncertainty, which is defined as one standard deviation.
• Figure 4: Scaling forms for the number density of areas from (a) positive and (b) negative signs in the SHE-Euler model. Corresponding outcomes from RSOS [($t_{\textrm{lab}} = 1600.5$; 51200.5), (a) negative (b) positive sign] and the ETC [($t_{\textrm{lab}} = 320.5$; $20480.5$), (a) positive (b) negative sign] are shown. Insets show the area-radius relation computed from clusters in the population of positive (a) and negative (b) signs in the SHE-Euler model. The convention is the one of the keys in the main panels. Dashed lines are guides-to-the-eyes with the indicated slopes.