Bernoulli clusters

Percolation Lab

Bond percolation: every vertex is present (filled squares). Each lattice edge is kept independently with probability p. Closed bonds stay visible as dim lines.

Square bondsquare cells, 4 neighboursno spanning cluster

Square bond · p = 0.000 · square cells, 4 neighbours

Empty bond configuration — every vertex present (filled squares). Closed bonds are not drawn, so the graph is disconnected. Drag p: each open bond appears in white.

open bondgiant clusterclosed bonds not drawn

p = 0 on ℤ² bond: the square cells, 4 neighbours. Filled marks are vertices (always present in bond percolation). Closed bonds are not drawn — nothing is connected yet. Drag p to see white open bonds appear.

Modelp is kept when you switch

Set p = 0 to see the empty lattice. Bond keeps every vertex; site shows only hollow vacancies. Honeycomb is hexagons, not a square grid.

Critical probability

p_c = 1/2

0.500000

subcritical

The square grid itself: four square cells meet at a vertex. All vertices stay; each of the four edges is kept independently with probability p.

Exact (Kesten, 1980). Self-dual.

Occupation p0.000 (-0.500 vs p_c)

0p_c1

Window size14 × 14

Scale n. Square and triangular use an n×n vertex window; honeycomb tiles about n×n hexagons.

Build redraws the uniforms. Dragging p reuses them (monotonic coupling).

Color

Vertices (all present): 196
Open / total bonds: 0 / 364
Clusters: 196
Largest: 1
Spanning mass: 0
θ̂ (span / n): 0.000