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Percolation Probability vs Density

Run multiple simulations across different densities to empirically measure the percolation probability curve. Watch as the S-shaped sigmoid curve emerges, showing the sharp phase transition around 59% density. Below threshold: nearly 0% crossing. Above threshold: nearly 100% crossing.
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Theory:

Percolation probability P(p) follows a sigmoid curve:

• p < p_c: P ≈ 0 (subcritical)
• p = p_c: P ≈ 0.5 (critical)
• p > p_c: P ≈ 1 (supercritical)

Critical point p_c ≈ 0.59 for Von Neumann lattice