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.

• The paper identifies unique coarsening dynamics in 2D growing interfaces belonging to the Kardar-Parisi-Zhang (KPZ) universality class, showing that one cluster population grows exponentially while the other follows a power law.
• The exponential growth of the giant cluster is tied to the sign of the KPZ nonlinearity λ, with one population forming a giant cluster covering nearly half the system area.
• The analysis validates that the number densities of cluster areas exhibit distinct, universal, population-dependent scaling forms, differing from those of linear dynamics.

Context and motivation

Clusters defined by the sign of height fluctuations about the mean level have been characterized at equilibrium and in nonequilibrium steady states of two-dimensional (2D) directed interfaces, but their coarsening dynamics had not been studied. Almeida, Oliveira, Arenzon, and Cugliandolo address this gap by simulating three lattice growth models believed to belong to the Kardar–Parisi–Zhang (KPZ) universality class and analyzing the connected regions above or below the mean height $\langle h\rangle$ (2601.14025). The central findings are that sign configurations are statistically time-invariant under rescaling by a single growing length, that the largest cluster in one population grows exponentially fast rather than as a power law, and that all scaling functions are shared across models with different microscopic rules.

Models and methodology

The authors simulate three growth rules on $L\times L$ square lattices with periodic boundaries: the SHE-Euler rule derived from an Euler discretization of the stochastic heat equation with multiplicative noise (a KPZ discretization free from numerical instabilities), the restricted solid-on-solid (RSOS) model with restriction parameter $n=3$, and an etching model (ETC). All models are initialized flat and verified to obey Family–Vicsek dynamic scaling with KPZ exponents. Clusters are extracted via a Hoshen–Kopelman labeling algorithm applied to the sign field $s(\mathbf{r},t)=\mathrm{sgn}[h-\langle h\rangle]$. A methodological subtlety is handled explicitly: for integer-height models, observables oscillate deterministically with the fractional part of $\langle h\rangle$, so measurements are taken on the laboratory clock $t_{\mathrm{lab}}=\langle h\rangle$ at half-integer values, which restores clean scaling.

Two-point correlator and statistical time invariance

The sign-sign correlator $C(r,t)$ decays monotonically, and defining the correlation length through $C(\xi,t)\equiv 0.2$ yields the growth law $\xi(t)\sim t^{1/z}$ with $z\approx 1.61$, consistent with the best numerical estimates for 2+1-dimensional KPZ. In both the growth regime ($L\times L$0) and the steady state ($L\times L$1), data collapse onto master curves when plotted against $L\times L$2; these curves are identical for SHE-Euler, RSOS, and ETC within numerical precision, establishing universality. Notably, the growth-regime and steady-state scaling functions are distinct from each other and from the correlator of the height fluctuations themselves. In the steady state, pairs at $L\times L$3 become slightly anti-correlated ($L\times L$4), reflecting the coexistence of few macroscopic clusters of opposite sign.

Asymmetric growth of the largest clusters

The most striking result is a pronounced up-down asymmetry tied to the sign of the KPZ nonlinearity $L\times L$5. The cluster population whose sign is opposite to $L\times L$6 rapidly develops a giant percolating cluster occupying nearly half the system area, while clusters of sign equal to $L\times L$7 grow more slowly through attachment and detachment events. Quantitatively:

• Population without giant cluster: the area fraction $L\times L$8 grows monotonically toward a universal steady-state value $L\times L$9, shared by all three models. In the core of the growth regime, $n=3$0 with $n=3$1.
• Giant-cluster population: the area fraction rises unusually fast to about 0.47, then more slowly to a maximum near 0.50, before settling at 0.488–0.499 in the steady state. Crucially, the early regimes collapse under the scale $n=3$2, implying a growing length

$n=3$3

with microscopic time $n=3$4. This exponential law is proper to KPZ dynamics — it is absent when $n=3$5 (Edwards–Wilkinson case), where no giant cluster forms — and contrasts sharply with the power-law growth $n=3$6 found in standard coarsening systems such as Ising or voter models.

The exponential-in-$n=3$7 growth means the giant cluster's characteristic length exceeds the ordinary correlation length $n=3$8, so the two populations are governed by genuinely different scales. This is a strong claim: coarsening here is not described by a single growing length, breaking the usual self-similar picture of domain growth.

Number densities of cluster areas

The number densities exhibit population-dependent scaling forms. For the population without the giant cluster, areas scale with $n=3$9: the dimensionless density $s(\mathbf{r},t)=\mathrm{sgn}[h-\langle h\rangle]$0 versus $s(\mathbf{r},t)=\mathrm{sgn}[h-\langle h\rangle]$1 collapses over roughly ten decades in $s(\mathbf{r},t)=\mathrm{sgn}[h-\langle h\rangle]$2 and five decades in $s(\mathbf{r},t)=\mathrm{sgn}[h-\langle h\rangle]$3, with algebraic regimes $s(\mathbf{r},t)=\mathrm{sgn}[h-\langle h\rangle]$4 ($s(\mathbf{r},t)=\mathrm{sgn}[h-\langle h\rangle]$5) for $s(\mathbf{r},t)=\mathrm{sgn}[h-\langle h\rangle]$6 and $s(\mathbf{r},t)=\mathrm{sgn}[h-\langle h\rangle]$7 ($s(\mathbf{r},t)=\mathrm{sgn}[h-\langle h\rangle]$8) for $s(\mathbf{r},t)=\mathrm{sgn}[h-\langle h\rangle]$9, followed by exponential decay. The small-area exponent is consistent with the scaling argument $\langle h\rangle$0 obtained from the equilibrium relation $\langle h\rangle$1 combined with the exact KPZ constraint $\langle h\rangle$2; this value is close to the exponent $\langle h\rangle$3 of hole distributions in the critical percolation backbone, and clearly distinct from the Edwards–Wilkinson value $\langle h\rangle$4.

For the giant-cluster population, the relevant scale is $\langle h\rangle$5, which quickly saturates at $\langle h\rangle$6; plotting $\langle h\rangle$7 versus $\langle h\rangle$8 yields system-size-independent curves with $\langle h\rangle$9, $t_{\mathrm{lab}}=\langle h\rangle$0, at small $t_{\mathrm{lab}}=\langle h\rangle$1, a faster decay, and a peak at $t_{\mathrm{lab}}=\langle h\rangle$2 signaling the recurrent macroscopic cluster. Neither the scaling argument nor $t_{\mathrm{lab}}=\langle h\rangle$3-based rescaling accounts for $t_{\mathrm{lab}}=\langle h\rangle$4: the giant cluster has fractal dimension $t_{\mathrm{lab}}=\langle h\rangle$5 yet $t_{\mathrm{lab}}=\langle h\rangle$6, underscoring its atypical character. Both scaling functions are shared by RSOS and ETC, confirming universality.

Limitations and open questions

Several caveats bear directly on the results. First, the algebraic regime governed by $t_{\mathrm{lab}}=\langle h\rangle$7 is not fully resolved: the authors concede it may be a finite-size artifact preceding an exponential cutoff. Second, the scaling derivation of $t_{\mathrm{lab}}=\langle h\rangle$8 borrows the equilibrium contour-loop result $t_{\mathrm{lab}}=\langle h\rangle$9 and applies it to a nonequilibrium growing system; its success is empirical rather than derived. Third, no analytical explanation is offered for the exponential law $C(r,t)$0 or for why the giant cluster selects the sign opposite to $C(r,t)$1; the connection is established numerically across three models plus the $C(r,t)$2 control. Fourth, all results concern flat initial conditions on the torus; whether the same asymmetry and scaling forms hold for curved geometries or other initial conditions remains open. Finally, the authors note that experimental verification — for instance in turbulent liquid crystals, semiconductor thin films, or paper combustion fronts, where KPZ scaling is documented — has not been attempted.

Conclusion

This work establishes that the coarsening of threshold-level clusters in 2D KPZ-like growing interfaces displays statistical time invariance with model-independent scaling functions, but with a fundamental sign asymmetry inherited from the $C(r,t)$3 term: one population forms a giant cluster growing exponentially fast in time, while the other follows correlation-length-controlled power-law coarsening. The number densities possess distinct, universal, population-dependent scaling forms unlike those of linear (Edwards–Wilkinson) dynamics. These results add new universal signatures to the KPZ class and pose concrete open problems: deriving the exponential growth law analytically, resolving the status of $C(r,t)$4, and testing the predicted asymmetry in experimentally realizable growing interfaces.