percolation

Percolation theory studies the behavior of connected clusters in random media. It is used to model how a fluid, information, or influence may move through a porous material, a network, or other disordered systems. In the simplest models, each site (node) or bond (edge) of a lattice is independently occupied with probability p and empty otherwise. The central question is whether a connected path of occupied sites or bonds spans the system, enabling long-range connectivity.

Two common models are site percolation and bond percolation. In site percolation, occupancy is assigned to sites; in bond percolation, to edges. Lattice models are typical, but continuum percolation also exists, where objects such as discs are placed randomly and connect when they overlap.

As p increases, the system undergoes a phase transition at a critical probability p_c. For p < p_c, only finite clusters exist; for p > p_c, an infinite cluster appears in the infinite lattice, enabling percolation. In finite systems, percolation is defined by the presence of a cluster that connects opposite sides or spans the system. Near p_c, properties show scaling behavior governed by universal exponents.

Threshold values depend on dimensionality and lattice type. For a two-dimensional square lattice, p_c ≈ 0.5927 for site percolation and p_c = 0.5 for bond percolation. In three dimensions, site percolation on the simple cubic lattice is about 0.3116, and bond percolation about 0.2488. The study of critical exponents, fractal nature of clusters, and finite-size effects is central to the theory.

Applications span materials science, geology, epidemiology, and network science, where percolation concepts quantify thresholds for flow, connectivity, or spread. Related topics include random graphs, percolation thresholds, and directed percolation.