🔵 Percolation Theory
In site percolation each cell of a 2D grid is occupied independently with probability p. At a sharp critical threshold p_c ≈ 0.5927 (square lattice), a cluster of occupied sites first spans the entire lattice — a second-order phase transition analogous to magnetisation in the Ising model. Below p_c clusters are finite; above p_c a giant spanning cluster fills a finite fraction of the lattice. At p_c the spanning cluster is a fractal with dimension D ≈ 1.896. Bond percolation works similarly but on edges; p_c ≈ 0.5 for the square lattice. Clusters are found by union-find (Hoshen-Kopelman). 🇺🇦 Українська
Type
Applications
Percolation models appear in many contexts: flow of oil through porous rock, spread of forest fires (each tree ignites neighbours with probability p), epidemic spreading on contact networks, conductivity of random resistor networks, and polymer gel formation. The universality hypothesis states that all 2D percolation systems belong to the same universality class regardless of the lattice geometry — only the value of p_c changes.