Critical Percolation Probabilities for the Next-Nearest-Neighboring Site Problems on Sierpinski Carpets

TL;DR

This study calculates the critical percolation probabilities for next-nearest-neighboring sites on Sierpinski carpets, revealing relationships with fractal dimension and connectivity, and aligning with known probabilities on square lattices.

Contribution

It introduces a method to compute percolation thresholds on fractal structures and establishes a relation among critical probability, fractal dimension, and connectivity.

Findings

Critical probabilities depend on fractal dimension D.

Relation among Pc, D, and Q established.

Results agree with known lattice probabilities as D approaches 2.

Abstract

In this paper, we compute the next-nearest-neighboring site percolation (Connections exist not only between nearest-neighboring sites, but also between next-nearest-neighboring sites.) probabilities Pc on the two-dimensional Sierpinski carpets, using the translational-dilation method and Monte Carlo technique. We obtain a relation among Pc, fractal dimensionality D and connectivity Q. For the family of carpets with central cutouts,, where, the critical percolation probability for the next-nearest-neighboring site problem on square lattice. As D reaches 2, which is in agreement with the critical percolation probability on 2-d square lattices with next-nearest-neighboring interactions.