Phase transitions in the node, edge, bootstrap, and diffusion percolation models on the Sierpiński carpet

Abstract

We investigate four types of percolation models — node, edge, bootstrap, and diffusion percolation — in three fractal graphs constructed on the Sierpiński carpet, employing the Monte Carlo method based on the Newman–Ziff algorithm. For each case, we calculate the percolation threshold and critical exponents ($\nu$, $\gamma$, and $\beta$) through the crossing of percolation probabilities and the finite-size scaling analysis, incorporating correction-to-scaling effects. Our results reveal that critical exponents of the percolation phase transition in the three fractal graphs exhibit universality across all four percolation models. Furthermore, we demonstrate that the hyperscaling relation $d\nu =\gamma +2\beta$ is also valid in the percolation phase transition on the Sierpiński carpet if the spatial dimension $d$ is replaced by the Hausdorff dimension.