PERCOLATE.

Site percolation · square lattice

Percolate The instant chance decides to connect

Open each cell of a grid with probability p and colour what touches what. Below the critical value the field is all islands. Cross pₒ ≈ 0.5927 and a single cluster leaps edge to edge — the whole lattice connecting in one step you can watch arrive.

Occupation probability
0.605 Islands only
Open sites 60.5%
Largest cluster —%
Clusters —
0.400.72

◄ ► raise / lower p · drag the rail

Reading the transition

Connection is not gradual. It arrives.

01 — The ruleTake a large square grid. Independently, make each cell open with probability p and closed otherwise. Two open cells belong to the same cluster if you can walk between them stepping only up, down, left or right through open neighbours.

02 — The thresholdFor low p you get scattered islands. Raise p and they grow and merge. On the infinite square lattice there is one sharp value, pₒ ≈ 0.5927, below which every cluster is finite and above which an infinite spanning cluster exists. Nothing warns you it is coming.

03 — Phase transitionThe largest cluster is the order parameter. It sits near zero, then rises almost vertically at pₒ — a genuine phase transition, the same mathematics behind a magnet losing its field or a gel setting. Chance, piled deep, becomes a law with an edge.

04 — Why it mattersPercolation decides when a forest fire jumps a firebreak, when a porous rock lets oil through, when a random network of routers stays reachable. The question is never how much is open — it is whether you have crossed the line.

Largest cluster · P∞(p)pₒ 0.5927
0 pₒ 1 1 0

The fraction of the lattice held by its largest cluster. Flat, then a cliff at pₒ. On a finite grid the cliff is a steep ramp; on the infinite lattice it is a true discontinuity in slope.

LatticeSite pₒBond pₒNote
Square 0.59274 0.50000 The grid on screen. Bond threshold is exactly one half.
Triangular 0.50000 0.34729 More neighbours connect sooner — site threshold is exactly half.
Honeycomb 0.69704 0.65271 Only three neighbours per site — connection comes late.
Cubic (3D) 0.31160 0.24812 A third dimension opens far more paths — spanning is cheap.