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

This paper utilizes Monte Carlo simulations to determine the critical thresholds for site and bond percolation on the Smith Hat aperiodic monotile, reporting values of approximately 0.8227 for site percolation, 0.7982 for bond percolation, and 0.5442 for site percolation on the dual graph.

The Gist

Imagine you are trying to build a massive, infinite road network across a strange, new continent. This continent isn't built with a neat grid of squares (like a city block) or a honeycomb pattern (like a beehive). Instead, it's built using a single, weirdly shaped tile called the "Smith Hat."

This "Hat" tile is a mathematical marvel discovered in 2023. It's the first shape ever found that can cover an entire floor without ever repeating the same pattern twice. It's like a puzzle that goes on forever but never looks the same way twice.

This paper is about figuring out how fragile or how strong this strange, non-repeating world is.

The Game: "The Flood"

To test the strength of this world, the researchers played a game they call Percolation.

Imagine it starts raining on this Hat-tile continent.

• The Tiles (or the Roads): Each tile (or the road connecting them) has a chance of being "open" (dry) or "closed" (flooded).
• The Goal: We want to know: How much rain can we have before the entire continent gets cut off?

If the rain is light, you can still drive from the left side of the map to the right side, or from the top to the bottom. But if the rain gets too heavy, the "flooded" spots will block every possible path. The moment the water rises high enough to stop all travel is called the Critical Threshold.

The Two Ways to Flood the World

The researchers tested two different ways the "flood" could happen:

1. The "Tile" Game (Site Percolation): Imagine the rain falls directly on the tiles. If a tile gets wet, it's closed. Can you still hop from one dry tile to another to cross the map?
2. The "Road" Game (Bond Percolation): Imagine the tiles are safe, but the roads connecting them are getting flooded. If a road is underwater, you can't cross it. Can you still find a dry path?

The Results: A Very High Barrier

The researchers ran millions of computer simulations (like playing the flood game over and over again with different random patterns) to find the exact tipping point.

Here is what they found for the Smith Hat tile:

• For the "Road" Game: You need to flood about 80% of the roads before the map gets cut off.
• For the "Tile" Game: You need to flood about 82% of the tiles before the map gets cut off.

Why is this surprising?
In normal, repeating patterns (like a city grid), you only need to flood about 50% of the roads to cut off the city. The Smith Hat world is much harder to cut off!

The Analogy:
Think of a standard city grid as a spiderweb. If you cut half the threads, the web falls apart.
The Smith Hat world is like a fortress made of interlocking stones. Because the shapes are so unique and fit together in complex, non-repeating ways, the "roads" are redundant. Even if you block a huge number of them, there are always weird, winding detours you can take to get across. It takes a lot of damage to break the connection.

Why Does This Matter?

You might ask, "Who cares about a weird hat-shaped tile?"

1. Quasicrystals: Real-world materials called "quasicrystals" (used in some non-stick pans and high-tech alloys) have atomic structures that look like these aperiodic tiles. Knowing how "connected" they are helps engineers understand how electricity or heat moves through them.
2. Fault-Tolerant Networks: If you are designing a computer network or a power grid that needs to survive random failures, the Smith Hat structure teaches us that complex, non-repeating patterns are incredibly robust. They can lose a huge chunk of their parts and still stay connected.

How Did They Do It?

Since you can't build an infinite floor of Smith Hats in your living room, they used Monte Carlo Simulations.

• They built a giant digital patch of these tiles on a computer.
• They randomly "flooded" parts of it thousands of times.
• They watched exactly when the path from one side to the other disappeared.
• They used math to guess what would happen if the map were truly infinite.

The Bottom Line

This paper is the first time anyone has calculated exactly how "connected" this new, weird shape is. They discovered that the Smith Hat tile creates a world that is surprisingly resilient. It takes a massive amount of damage to break the flow, proving that sometimes, breaking the rules of repetition (aperiodicity) creates a structure that is stronger than the standard, repeating patterns we are used to.

In short: The Smith Hat tile is a master of "keeping the lines open," even when things are falling apart around it.

Technical Summary

1. Problem Statement

The paper addresses a gap in the statistical physics and geometry literature regarding the Smith Hat tile, the first known aperiodic monotile (discovered in 2023). While the geometric properties of the Smith Hat tiling are established, its percolation critical thresholds ($p_c$) had not been determined.

The core problem is to calculate the critical probability $p_c$ for both site percolation (vertices open/closed) and bond percolation (edges open/closed) on the infinite Smith Hat tiling. Unlike periodic lattices (e.g., square or triangular), the Smith Hat tiling lacks translational symmetry, meaning local environments (patches) vary, making analytical derivation of $p_c$ intractable. The authors aim to establish these thresholds using numerical simulation to understand how the unique "hat" geometry influences global connectivity and phase transitions.

2. Methodology

The authors employed Monte Carlo simulations combined with Finite-Size Scaling (FSS) analysis to estimate the critical thresholds for an infinite system.

A. Lattice Generation

• Substitution Rule: The tiling is generated using the substitution rules defined by Smith et al. (2024). The process involves grouping "hat" prototiles into four metatiles (H, T, P, F) and iteratively inflating and subdividing them.
• Graph Construction: Two distinct graph structures were constructed for the simulations:
  1. Edge Percolation: Vertices of the tiling are nodes; edges of the tiling are connections.
  2. Tile (Dual) Percolation: Each hat tile is a node; edges exist between adjacent tiles.
• Implementation: The generation logic was adapted from the original JavaScript implementation to Python. A KD-tree was used to merge coincident vertices with a tolerance of $\epsilon = 10^{-5}$ to ensure topological accuracy.

B. Simulation Protocol

• System Sizes: Simulations were run on square frames of side length $L$ ranging from 10 to 400, centered at coordinate $(200, -100)$ to maximize sampling within the generated patch.
• Percolation Types:
  • Site Percolation: Sites are opened sequentially in a random order until a spanning cluster forms.
  • Bond Percolation: Edges are opened sequentially.
• Estimators: Four percolation criteria were tracked for each trial:
  • $p^R_L$: Rightward spanning.
  • $p^D_L$: Downward spanning.
  • $p^I_L$: Intersection (both Right and Down).
  • $p^U_L$: Union (either Right or Down).
  • $p^A_L$: Average ($p^A_L = \frac{1}{2}(p^R_L + p^D_L)$), justified by the hypothesis that the Smith Hat tiling is isotropic at large scales.
• Statistical Analysis:
  • Trials: 1,000 independent experiments per system size $L$.
  • Extrapolation: The critical probability $p_c(L)$ was extrapolated to $L \to \infty$ using the scaling law:
    $$p_c(L) = p_c + A_X L^{-1/\nu}$$
  • Critical Exponent: The authors assumed the universal 2D critical exponent $\nu = 4/3$, consistent with periodic lattices and previous work on Penrose tilings.
  • Regression: A Weighted Least Squares (WLS) linear regression was performed on $p_c(L)$ vs. $L^{-3/4}$ to determine the intercept ($p_c$) and its confidence interval, utilizing the $t$-distribution for small sample robustness.

3. Key Contributions

1. First Determination of $p_c$: This is the first study to establish the percolation thresholds for the Smith Hat monotile.
2. Dual Graph Analysis: The paper distinguishes between "edge percolation" (vertex-based) and "tile percolation" (dual graph), providing values for both.
3. Validation of Methodology: The authors verified their algorithm and pipeline against known analytical thresholds for the Square Lattice, Triangular Lattice, and Penrose P3 tiling, demonstrating high precision and reliability.
4. Isotropy Assumption: The work provides empirical support for the hypothesis that the Smith Hat tiling behaves as an isotropic elastic continuum at large scales, allowing the use of averaged estimators.

4. Results

The study yielded the following critical probabilities (with 95% Confidence Intervals):

Percolation Type	Critical Probability ($p_c$)	95% Confidence Interval
Site Percolation (Edge)	0.822725	[0.822636, 0.822815]
Bond Percolation (Edge)	0.798161	[0.798073, 0.798250]
Site Percolation (Tile/Dual)	0.544247	[0.544044, 0.544450]
Bond Percolation (Tile/Dual)	0.201839	[0.201750, 0.201927]

Note: The Tile Bond threshold is derived via duality ($1 - p_{bond}^{edge}$).

Key Observations:

• High Thresholds: The edge percolation thresholds ($p_c \approx 0.82$ for site, $\approx 0.80$ for bond) are significantly higher than those of standard 2D periodic lattices (e.g., Square site $p_c \approx 0.59$, Triangular site $p_c = 0.5$).
• Coordination Number Correlation: The authors attribute the high thresholds to the low average coordination number of the Smith Hat tiling (approx. 2.31). Lower connectivity requires a higher occupation probability to form a spanning cluster.
• Convergence: The intersection ($p^I$) and union ($p^U$) estimators converged to the same value as $L \to \infty$, confirming the isotropic nature of the tiling.

5. Significance and Implications

• Benchmark for Aperiodic Monotiles: These results provide the first quantitative benchmarks for the geometric phase behavior of the Smith Hat tile, serving as a foundation for future analytical work on aperiodic structures.
• Universality Class: The results support the hypothesis that aperiodic tilings share the same universality class ($\nu = 4/3$) as periodic 2D lattices, despite the lack of translational symmetry.
• Physical Applications:
  • Fault Tolerance: The high percolation threshold implies that networks modeled by the Smith Hat geometry are highly robust; a large fraction of components must fail before global connectivity is lost.
  • Quasicrystals: As the Smith Hat tiling models quasicrystalline materials, these findings offer insights into transport properties and connectivity in materials with long-range order but no periodicity.
• Future Directions: The paper suggests extending this analysis to other members of the Hat tile family and rigorously proving the critical exponents for this specific geometry.

6. Limitations

• Precision: While confidence intervals are tight ($\pm 0.00005$), further precision is limited by the sequential nature of the current code and available compute time.
• Assumptions: The results rely on the assumption of isotropy and the universality of the critical exponent $\nu = 4/3$, which, while supported by precedent, has not been analytically proven for the Smith Hat geometry.
• Generalization: The results are specific to the $(1, \sqrt{3})$ variant of the Hat tile; other variants may exhibit different thresholds.