Large clusters in a correlated percolation model

TL;DR

The paper studies a correlated site percolation problem on a cubic lattice where sites are removed by an ${ m N}$-step random walk with ${ m N}=uL^3$, identifying a percolation threshold at $u_c = 3.15$. It develops and tests scaling relations for large clusters, showing the mean mass of the $r$th largest cluster scales as $M_r ∼ L^{d_f}/r^{d_f/d}$ with a fractal dimension $d_f = (d+2)/2 = 5/2$ in $d=3$, yielding $M_r ∼ L^{5/2}/r^{5/6}$. Numerical simulations using the Hoshen–Kopelman algorithm on sizes up to $L=512$ with $10^4$ samples per size corroborate the theory, giving $d_f obreak[=] 2.50$–$2.52$ and extrapolated slopes near $-5/6$, albeit with finite-size deviations due to a cutoff function. The work provides a practical framework for probing the incipient infinite cluster in correlated percolation by ranking large clusters, with implications for understanding connectivity in correlated media.

Abstract

We consider a correlated site percolation problem on a cubic lattice of size $L^3$, with $16\le L\le 512$. The sites of an initially full lattice are removed by a random walk of ${\cal N}=uL^3$ steps. When the parameter $u$ crosses a threshold $u_c=3.15$, a large system transitions between percolating and non-percolating states. We study the $L$-dependence of the mean mass (number of sites) $M_r$ of the $r$th largest cluster, as well as $r$-dependence of $M_r$ for various system sizes $L$ at $u_c$. We demonstrate that $M_r\sim L^{5/2}/r^{5/6}$ for moderate or large $L$ and $r\gg 1$, and also conclude that for {\em any} $r$ the fractal dimensions of the clusters are $5/2$.

Figures (3)

• Figure 1: Logarithmic plots of the mean mass $M_r$ of $r$th largest cluster at $u_c$, averaged over $10^4$ configurations, as a function of the system size $L$ ranging between $L=16$ and 512, for ranks (top to bottom) $r=1$, 2, 3 and 10. Statistical errors of the points are smaller than the symbol sizes. Slopes of the plots represent the fractal dimension $d_{\rm f}$.
• Figure 2: Logarithmic plots of the mean mass $M_r$ of $r$th largest cluster as a function of the rank $r$ for system sizes ranging (bottom-left to top-right) from $L=16$ to $L=512$. Each data point has been averaged over an ensemble of $10^4$ configurations, and statistical errors are significantly smaller than the symbol sizes. The discreteness of specific values of $M_r$ becomes apparent in the steps at the bottom of all graphs, when the ensemble averages cannot smooth-out the discrete (integer) values of masses in each specific realization. For very large $r$ there are no clusters ($M_r=0$) and the logarithm drops to $-\infty$. Dashed lines are the linear fits in the selected range of $r$ and $M_r$ values, and the numbers near them indicate their slopes. ($L=16$ has no valid range for the required fit, but it has a rather straight segment of slope $\approx -1.0$.)
• Figure 3: Slopes of the fits in Fig. \ref{['fig:Mvsr']} vs. $L^{-1/2}$, where $L$ is the linear system size. Errors in the slopes are smaller than the symbol sizes. The open circle represents the $L=16$ data which has no valid range for the fit. (See the caption of the Fig. \ref{['fig:Mvsr']}.) Only the slopes shown by the full circles represent the valid data, and are used in the extrapolation $L\to\infty$. The arrow indicates the extrapolated value.