Percolation Phase Transition Lab: Sharp Thresholds & Critical Universality

Lattice Connectivity (Square Bond, L = 40) Click to toggle site/bond

Percolation Probability P_∞(p) & Critical Singularity p = 0.50 | P_inf = 0.00

In the infinite lattice limit (L → ∞), P_∞(p) = 0 for p < p_c, and jumps sharply following the power law (p - p_c)^β where β = 5/36 ≈ 0.1389 in 2D.

Occupation Probability (p)

Critical parameter p: 0.500

▲ 0.50 ▲ 0.593 ▲ 0.50 ▲ 0.696

Lattice Geometry:

Square Bond (p_c=0.5)

Square Site (p_c≈0.593)

Triangular Site (p_c=0.5)

Honeycomb Site (p_c≈0.696)

Physical Scenarios:

Coal Porosity (Gas)

Forest Wildfire

Viral Transmission

Cluster Telemetry

Spanning Status

Giant Cluster %

0.0%

Total Clusters

Avg Cluster Size ⟨s⟩

0.0

Algorithm: Weighted Quick-Union with Path Compression (Tarjan disjoint-set). Computes connected components across boundaries in O(N α(N)) time.

Rigorous Mathematical Status: Proved Theorems vs. Open Conjectures

✓ Landmark Proved Theorems

Kesten's Critical Probability Theorem (1980)

Harry Kesten · Square Lattice Bond Percolation

Rigorously proved that p_c = 1/2 for bond percolation on ℤ². Settled decades of heuristic arguments by demonstrating that the Russo-Seymour-Welsh (RSW) lemma and self-duality enforce exactly p_c = 1/2.

Sharpness of the Phase Transition (1986–2016)

Menshikov (1986), Aizenman-Barsky (1987), Duminil-Copin & Tassion (2016)

Proved exponential decay of cluster sizes below p_c (P(0 ↔ x) ≤ e^-c|x| for p < p_c) across all dimensions d ≥ 2, establishing that the transition is strictly sharp with no intermediate regime.

Conformal Invariance & Cardy's Formula (2001)

Stanislav Smirnov (Fields Medal 2010)

Proved Cardy's crossing formula on triangular site percolation, demonstrating that the scaling limit converges to Schramm-Loewner Evolution (SLE₆). Derived the exact 2D critical exponents β = 5/36, γ = 43/18.

? Unresolved Open Conjectures

Exact Value of p_c for 2D Square Site Percolation

Open Problem in Discrete Probability

Despite being 2D, square site percolation lacks planar self-duality. Numerically computed to high precision as p_c ≈ 0.59274605, yet no closed-form algebraic expression exists and its exact nature remains unproved.

Universality Across All 3D Geometries

Conjectured Critical Exponents in d = 3

Physicists predict universal critical exponents (β ≈ 0.41, ν ≈ 0.875) for all 3D lattices (cubic, BCC, FCC, random voronoi). A mathematically rigorous proof of 3D universality remains entirely open.

Continuity of the Phase Transition in d = 3 (P_∞(p_c) = 0)

Open for 3D Lattices

While proved for ℤ² (by Harris) and for high dimensions d ≥ 11 (Hara-Slade), whether an infinite cluster can exist precisely at the critical threshold p_c in 3D remains an unsolved open problem.