Each cell of a square lattice is independently occupied with probability p. Below a critical p the occupied cells form only small, isolated clusters; above it a single cluster spans the whole system. This is called an infinite spanning cluster. Watch the giant cluster appear as you sweep p across the percolation threshold (~0.5927.. for site percolation on a square lattice).
How it works: Start runs a repeated sweep over site/edge occupation probabilities p. The graphs average over all cycles and demonstrate convergence to theoretically predicted values as the simulation progresses. The lattice shown is sampled at exactly the given p and the orange cluster, when present, is the one spanning top to bottom, and the white line traced through it is the shortest crossing (lattice view is cropped by visual elements).
How it works
Start runs a repeated sweep over site/edge occupation probabilities p.
The graphs average over all cycles and demonstrate convergence to theoretically predicted values as the simulation progresses.
The lattice shown is sampled at exactly the given p and
the orange cluster, when present, is the one spanning top to bottom, and the
white line traced through it is the shortest crossing.
Sites or Edges:
Site percolation occupies each site on the lattice with probability p
and joins occupied neighbours. Edge percolation keeps every site
and opens the edges between them with probability p. They are different
models with different critical percolation thresholds, with the threshold for edge percolation being exact pc = 1/2[2],
against 0.5927… for site percolation, a numerically calculated value[3].
Graphs:
(a) mean finite cluster size S(p); Average cluster size (# of included sites/edges) of all finite clusters. Infinite clusters
are not included in the calculation. At probabilities higher than the critical threshold Pc [orange dashed line], average finite clusters not
included in spanning cluster. (b) cluster-size distribution ns at the given probability p (log-log, ~power law at pc);
There exists large amounts of small clusters and few large clusters. (c) percolation strength P∞(p), the fraction of sites included the infinite spanning cluster.
Below the critical threshold, since there is no spanning cluster, the value is 0; (d) spanning probability Π(p), the fraction of sampled states that percolate / contain a spanning cluster. The orange dashed line is the
theoretical (or numerically calculated) critical threshold value Pc; (e) the shortest crossing $\ell/L$, averaged over the states that span[4]. Below
pc there exists few spanning clusters which are resultant of the finite size of the lattice used for the simulation,
so the data starts where spanning clusters begin to appear in the simulation runs.
