Square lattice occupation p: clusters, left–right spanning, p_c ≈ 0.593; rough box-counting D̂ on the largest cluster.
About this model
Site percolation on a square lattice colors each cell independently “open” with probability p and “closed” otherwise. Open cells that share an edge form clusters. Near the critical threshold p_c ≈ 0.592746 on the infinite square lattice, correlation lengths diverge and the spanning cluster (when it exists) is a fractal with fractal dimension D_f = 91/48 in two dimensions. This simulator draws one finite realization at your chosen p, labels connected components, highlights the largest cluster, and reports whether any cluster connects the left and right edges (spanning need not be the largest cluster). A crude box-counting slope on the largest cluster gives a rough D̂ when the cluster is large enough; it is noisy for small systems and should be read as illustrative.
Who it's for: Students learning phase transitions on networks, scaling, and universality in statistical physics.
Key terms
Percolation
Critical probability p_c
Spanning cluster
Fractal dimension
Box counting
Correlation length
How it works
Square-lattice site percolation: random occupation, cluster labeling, spanning probability near p_c, and a crude fractal dimension from box counting on the largest cluster.
Frequently asked questions
Why does my estimated D̂ differ from 91/48?
91/48 describes the spanning cluster at criticality in the thermodynamic limit. A single L×L sample is finite, sub-leading corrections matter, and box counting with a handful of box sizes is statistically noisy. Expect qualitative agreement near p_c only when the largest cluster is big and you average many realizations.
Is this bond or site percolation?
Site percolation: vertices (cells) are occupied randomly; edges are implicit between occupied nearest neighbors. Bond percolation would randomize links instead; the critical probability is different.
Does “spanning” require the highlighted largest cluster to cross the system?
No. The gold frame means some open cluster connects the left column to the right column. The largest cluster is highlighted for visualization and for the box-counting estimate D̂, but it might sit in the interior while a thinner cluster provides connectivity.