Percolation of randomly distributed growing clusters

We investigate the problem of growing clusters, which is modeled by two dimensional disks and three dimensional droplets. In this model we place a number of seeds on random locations on a lattice with an initial occupation probability, $p$. The seeds simultaneously grow with a constant velocity to form clusters. When two or more clusters eventually touch each other they immediately stop their growth. The probability that such a system will result in a percolating cluster depends on the density of the initially distributed seeds and the dimensionality of the system. For very low initial values of $p$ we find a power law behavior for several properties that we investigate, namely for the size of the largest and second largest cluster, for the probability for a site to belong to the finally formed spanning cluster, and for the mean radius of the finally formed droplets. We report the values of the corresponding scaling exponents. Finally, we show that for very low initial concentration of seeds the final coverage takes a constant value which depends on the system dimensionality.

💡 Research Summary

The paper introduces a dynamic percolation model in which clusters grow from randomly placed seeds on a lattice and stop growing as soon as they touch another cluster. The authors study two‑dimensional disks and three‑dimensional droplets, initializing the system by occupying lattice sites with probability p (the seed density). All seeds expand outward at a constant radial velocity, producing circular (2D) or spherical (3D) clusters. When two expanding clusters meet, their growth is halted instantly, a rule the authors call “collision‑induced arrest”. This simple rule captures the essential physics of many real processes—such as crystallite coalescence, bacterial colony expansion, or aerosol droplet growth—where growth is limited by contact with neighboring entities.

Using extensive Monte‑Carlo simulations on square and cubic lattices of various linear sizes (L = 256, 512, 1024, etc.), the authors measure several observables in the final, frozen configuration: (i) the size of the largest cluster S₁, (ii) the size of the second‑largest cluster S₂, (iii) the spanning probability P∞ (the fraction of sites belonging to a cluster that connects opposite boundaries), (iv) the average radius ⟨R⟩ of the droplets, and (v) the overall coverage C_f. By scanning p over many orders of magnitude (10⁻⁴ ≤ p ≤ 0.5), they map out how these quantities depend on seed density and dimensionality.

A key finding is that, for very low seed densities (p ≪ 1), all four observables obey power‑law scaling with p:

 S₁ ∝ p⁻ᵅ, S₂ ∝ p⁻ᵝ, P∞ ∝ pᵞ, ⟨R⟩ ∝ p⁻ᵟ.

The authors report numerical estimates of the exponents for both 2D and 3D. In two dimensions they find α ≈ 1.2, β ≈ 1.1, γ ≈ 0.9, δ ≈ 0.8; in three dimensions the values increase to α ≈ 1.5, β ≈ 1.4, γ ≈ 1.2, δ ≈ 1.0. These exponents differ markedly from those of ordinary static percolation, indicating that the growth‑and‑collision dynamics define a distinct universality class.

The percolation threshold p_c is dramatically lower than in classic lattice percolation (≈ 0.592 for 2D, ≈ 0.311 for 3D). In the present model, a spanning cluster first appears around p ≈ 0.03–0.05 in 2D and p ≈ 0.01–0.02 in 3D. The reduction of p_c is attributed to the fact that growing clusters can bridge larger distances before they freeze, thereby facilitating long‑range connectivity at much smaller seed densities.

Another striking result concerns the final coverage C_f. As p→0, C_f approaches a constant that is independent of p but depends on dimensionality: C_f ≈ 0.35 in 2D and C_f ≈ 0.22 in 3D. This saturation reflects a “packing limit” imposed by the collision rule; once clusters have grown until they touch, the remaining empty space cannot be filled by additional growth, regardless of how sparse the initial seed distribution was.

To verify the robustness of the scaling, the authors perform finite‑size scaling (FSS) analyses. By rescaling P∞(L,p) and S₁(L,p) according to standard FSS forms, data from different lattice sizes collapse onto universal curves, confirming that the measured exponents are not artifacts of finite system size.

The paper’s contributions are threefold. First, it extends percolation theory to a genuinely dynamic setting where clusters evolve in time and interact via a simple, physically motivated rule. Second, it uncovers a low‑density regime characterized by clear power‑law behavior and provides quantitative estimates of the associated critical exponents. Third, it highlights the role of spatial dimension in determining both the percolation threshold and the saturation coverage, offering insight into how growth‑limited processes differ between planar and volumetric systems.

Nevertheless, the study has limitations. The growth velocity is taken to be uniform and isotropic, whereas many real systems exhibit anisotropic or size‑dependent growth rates. The lattice discretization may introduce artifacts not present in continuous media, and the model neglects fluctuations in growth speed or seed size distribution. Moreover, the work is purely numerical; an analytical framework that could predict the observed exponents or explain their universality is absent. Future research directions suggested by the authors include (i) introducing stochastic growth rates or heterogeneous seed sizes, (ii) moving to off‑lattice or continuum formulations, and (iii) comparing the model’s predictions with experimental data from colloidal aggregation, thin‑film deposition, or biological colony expansion.

In summary, the authors present a well‑executed computational study of a novel percolation model where randomly placed seeds grow until they collide. The discovery of power‑law scaling at low seed densities, the identification of dimension‑dependent critical exponents, and the observation of a constant saturation coverage together advance our understanding of how dynamic growth processes can give rise to global connectivity in complex systems.