Site percolation simulation

Is there a sharp threshold at which random sites suddenly connect across a whole lattice?

▶ Run the simulationSee the measured result

Units: dimensionless occupation probability (square-lattice site-percolation threshold; no closed form — Jacobsen 2015 graph-polynomial value 0.59274605079210(2))

How the lab tests it

Occupy each site of a square lattice independently with probability p; find connected clusters with union-find. Sweep p and, over many random lattices at each value, measure the spanning probability Π(p) (a cluster reaching top↔bottom) and the largest-cluster fraction (the order parameter); locate p_c at the Π = ½ crossing.

What it checks

a connectivity phase transition at the known site-percolation threshold p_c ≈ 0.5927 for the square lattice — below it only finite islands, above it a lattice-spanning 'infinite' cluster