Percolation Critical Probability of Aperiodic Smith Hat tile(1, $\sqrt{3}$)

Abstract

The Smith Hat tile is the first known aperiodic monotile, having been discovered in 2023. The simple structure, constructed using only 8 kites, is unique and well motivated for analysis within percolation theory. The primary goal of this paper is to discover the critical threshold $p_c$ in both site and bond Bernoulli structures using Monte Carlo simulation for the Smith hat tile(1,$\sqrt{3}$). Our findings are site and bond values of $p_c^s=0.822725\pm 0.000044$ and $p_c^b=0.798161\pm 0.000044$ for edge percolation and $0.544247\pm 0.000101$ for site percolation on the dual graph.

arXiv comment18 pages, 10 figures