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

TL;DR

This paper determines the percolation critical probabilities for the first known aperiodic monotile, the Smith Hat tile, using Monte Carlo simulations in both site and bond percolation models.

Contribution

It provides the first known estimates of critical thresholds for the Smith Hat tile in percolation theory, a newly discovered aperiodic monotile.

Findings

Site percolation threshold: 0.822725 ± 0.000044

Bond percolation threshold: 0.798161 ± 0.000044

Dual graph site percolation threshold: 0.544247 ± 0.000101

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.