Percolation on multifractal, scale-free weighted planar stochastic porous lattice

Abstract

We introduce the Weighted Planar Stochastic Porous Lattice (WPSPL), a geometrically disordered substrate generated by iteratively subdividing a unit square. At each step a block is selected with probability proportional to its area, divided into four parts, and one sub-block is retained (removed) with probability $q$ ($1-q$). We show analytically that the WPSPL exhibits multifractality for each of its infinitely many nontrivial conserved quantities and demonstrate numerically that its snapshots at different times are statistically self-similar. The dual of the lattice forms a complex network with a power-law degree distribution. Motivated by these properties of this porous lattice, we study bond percolation on the WPSPL, determine the percolation threshold, and estimate the critical exponents $α$, $β$, and $γ$ associated with the specific heat, order parameter, and susceptibility, respectively. The exponents vary continuously with $q$, reflecting a family of distinct universality classes as the global dimension of the lattice depends on $q$. Remarkably, the Rushbrooke inequality, $α+ 2β+ γ\ge 2$, is satisfied in near equality. Notably, the nonporous case ($q=1$) has a global dimension $2$ but lies outside the universality class of conventional two-dimensional lattices. Our results highlight how geometric disorder, multifractality, scale-free coordination number disorder, and porosity produce unconventional critical behavior.

Figures (10)

• Figure 1: A snapshot of the stochastic for p=0.95 and t=5000. The shaded cells indicate that the cells were deleted.
• Figure 2: The $f(\alpha(k))$ spectra for (a) $q = 0.95$ and (b) $q = 0.85$, shown for $m = 1,\ 1.5,\ 3,$ and $5$ in each case. In all plots, the maximum occurs at $k = 0$, which corresponds to the fractal dimension of the support, in agreement with the theoretical prediction.
• Figure 3: The natural logarithm of the area distribution function $C(a,t)$ plotted as a function of $a$ for $t = 5000,\ 10{,}000,\ 15{,}000,$ and $30{,}000$: (a) $q = 0.95$ and (b) $q = 0.85$. In both cases, the plots are approximately linear in the tail region, indicating an exponential form of the distribution. The same data as in Figs. (\ref{['fig:3a']}) and (\ref{['fig:3b']}) are used to plot the dimensionless quantities $\ln(C(a,t),t^{-\sqrt{3+q}})$ versus $a t$ in (c) for $q=0.95$ and in (d) for $q=0.85$. An excellent collapse of the distinct curves shown in Fig. (\ref{['fig:3']}) is observed.
• Figure 4: The degree distribution $P(k)$ for the dual of the WPSPL network is shown on a log--log scale in (a) for $q=0.85$ and in (b) for $q=0.95$. The data points are averages over $50000$ independent realizations. In both cases, the distributions are approximately linear, indicating a power-law behavior. However, the heavy tail implies poor statistics for large $k$, making a direct estimation of the exponent from $P(k)$ unreliable. To mitigate this problem, the insets display the cumulative degree distribution $P(k' > k)$ computed from the same data as it remove the fat tail and provides a more reliable estimate of the scaling exponent. The exponent of $P(k)$ is then obtained by adding one to the exponent measured from the cumulative distribution.
• Figure 5: Spanning probability $W(p,L)$ as a function of $p$ for bond percolation on the WPSPL is shown for different system sizes at block-retaining probabilities (a) $q = 0.90$ and (b) $q = 0.85$. From the simulations, the estimated percolation thresholds are $p_c = 0.389389$ for $q = 0.90$ and $p_c = 0.416496$ for $q = 0.85$. Panels (c) and (d) show plots of $\log(p_c - p)$ versus $\log L$ for $q = 0.90$ and $q = 0.85$, respectively. The resulting slopes yield the correlation-length exponents $1/\nu = 0.538 \pm 0.002$ for $q = 0.90$ and $1/\nu = 0.487 \pm 0.001$ for $q = 0.85$.