Recursive percolation

We introduce a simple lattice model in which percolation is constructed on top of critical percolation clusters, and find compelling numerical evidence that it can be repeated recursively any number $n$ of generations. In two dimensions, we determine the percolation thresholds up to $n=5$. The corresponding critical clusters become more and more compact as $n$ increases, and define universal scaling functions of the standard two-dimensional form and critical exponents that are distinct for any $n$. This family of exponents differs from previously known universality classes, and cannot be accommodated by existing analytical methods. We confirm that recursive percolation is well defined also in three dimensions.

Figure 1

Wrapping probability $R_2^n$ vs $p^n$ for $n=1,2,3,4$. The superscript $n$ is for the $n\mathrm{th}$ generation. Percolation threshold is located by the approximately common crossing for different sizes $L$, as denoted by different colors.

Figure 2

Illustration of recursive single-cluster growing processes. The $n=0,1,2$ bonds and clusters are marked in gray, green, and black, respectively.

Figure 3

Probability distribution $P(s,L)$ in the single-cluster growing procedure. The size $L=16384$ in the main plot. The red ($n=0$) and blue ($n=1$) data are for $p^0=p_{\mathrm{c}}^0=1/2$ and $p^1=p_{\mathrm{c}}^1=0.654902$, respectively. The black curve with $p^1=0.6<p_c^1$ displays subcritical behavior. The red/blue straight lines have slopes $1-\tau \equiv -d/d_{\mathrm{F}}^n$, with $d_{\mathrm{F}}^0=91/48$ and $d_{\mathrm{F}}^1=1.8573$. The right-top inset shows the product $sP(s,L)$, and the left-bottom corner displays $s^{\tau -1}P(s,L)$ for $n=1$ vs $s/d_{\mathrm{F}}^1$, with $L=256,512,...,16384$.

Figure 4

Fractal dimensions. (a) Effective dimensions $d_{\mathrm{H}}^1\left(L\right)$ vs the variable $u=\left(p^1-p_{\mathrm{c}}^1\right)L^{y_t^1}$; $d_{\mathrm{H}}^1\left(L\right)$ approaches the exact value $4/3$ for $u\gg 0$. (b)–(d) The critical exponent $d_{\mathrm{X}}^n$ vs generation $n$, with $\text{X}=\text{H},\text{F},\text{R}$, respectively. The “$\times$” points are from the fitting results. For each of the exponents, the $n$ dependence can be fitted to a common $n\to \infty$ limit, as shown.