An interactive study of the phase transition in two-dimensional bond percolation. Adjust the bond probability and lattice size, then generate a sample to see how connected clusters form.
Bond percolation models the formation of connected clusters on a lattice in which each edge between neighbouring sites is present, independently, with probability p. The resulting random graph admits a startlingly precise mathematical structure for so simple a definition.
On the two-dimensional square lattice, the system undergoes a sharp phase transition at the critical probability pc = ½. Below this threshold the lattice fragments into many small, disconnected components; above it, an infinite cluster emerges and spans the entire system. The visualisation above samples a finite realisation of this process and colours each connected component distinctly.