Watch the giant cluster emerge as site occupancy crosses pc ≈ 0.593.
Each color is a connected cluster; red/orange = largest cluster.
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Percolation Theory
Site percolation on a square lattice: each cell is occupied independently with
probability p. Occupied cells that share an edge form clusters.
The phase transition: below pc ≈ 0.59274,
all clusters are finite and their sizes decay exponentially. Above pc, a single
giant cluster emerges that spans the system — its size scales as
(p − pc)β with β = 5/36 ≈ 0.139.
At criticality: cluster sizes obey a power law; the largest cluster has
fractal dimension df = 91/48 ≈ 1.896. There is no exact closed-form expression
for pc on the square lattice — it is known only numerically to high precision.