Percolation Critical Probability of Aperiodic Smith Hat tile(1, $\sqrt3$)

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,$\sqrt3$). 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.