Percolation & Critical Phase Transitions Simulator

Historical Scenarios:

Lattice: Square Bond

Occupied Ratio: 50.2%

Spanning Fluid Flow: BLOCKED

Occupation Probability (p) 0.500

Lattice Geometry & Mode

Grid Resolution (L × L) 48×48

Injection Source Direction

Lattice Metrics & Clusters

Total Nodes/Sites

2,304

Disjoint Clusters

342

Largest Cluster Mass

612

Fraction of Lattice (P∞)

26.5%

The system shifts sharply at p = p_c. Below p_c, open paths stay bounded in tiny local islands. Above p_c, a macroscopic "infinite" cluster spans the medium.

Phase Transition Curve P(p) Live Sweep

p=0.0 (Dead) p_c Threshold p=1.0 (Full)

Orange curve indicates empirical probability of forming an end-to-end spanning path computed via fast Disjoint-Set union-find tests.

Physical Flow Context

1957 Coal Gas Mask Filter: Simon Broadbent and John Hammersley modeled how poisonous gas diffuses through labyrinthine porous coal blocks. Rather than particles moving randomly (diffusion), fluid passes through random stationary pores (percolation).

Mathematical Dossier: Proved Theorems vs. Open Conjectures

Decades of breakthroughs in rigorous probability theory from 1957 to the 2022 Fields Medal.

✔ Rigorously Proved Milestones

Harry Kesten (1980): Exact Bond Threshold on ℤ²

Proved analytically that for bond percolation on the square lattice, the critical threshold is exactly p_c = 1/2, resolving Broadbent and Hammersley's foundational question.

Harris (1960) & FKG Inequality (1971)

Established correlation inequalities showing that increasing occupancy strictly increases the likelihood of positive flow events (non-negative association).

Aizenman, Kesten & Newman (1987): Uniqueness of Infinite Cluster

Proved that for any d ≥ 2 and any p > p_c, there is almost surely exactly one unique infinite connected cluster.

Hugo Duminil-Copin (2022 Fields Medal): Sharpness of Phase Transitions

Demonstrated universal sharp threshold behavior across generic dependent percolation models and random-cluster systems using randomized algorithms and Boolean analysis.

⚡ Active Open Conjectures

Exact Value of p_c for 3D Cubic Lattice

While rigorously proved that 0.2488 < p_c(ℤ³) < 0.2489, no exact closed-form algebraic expression is known. 3D percolation remains notoriously resistant to 2D dualities.

Universality of Critical Exponents in 3 ≤ d ≤ 5

In d=2, exponents are known via Schramm-Loewner Evolution (SLE); in d ≥ 6, mean-field theory holds. The intermediate spatial dimensions (d=3, 4, 5) remain unproven non-perturbative mysteries.

Continuum Conformal Invariance in 3D

Stanislav Smirnov (2001) proved conformal invariance of critical site percolation on the 2D triangular grid. Proving or disproving analogous spatial symmetries in d=3 is completely open.