Percolation Phase Transition

About Percolation Phase Transition

Percolation theory studies the emergence of connected clusters in random lattices. On a 2D square lattice, each site is independently occupied with probability p. Below the critical threshold p_c (approximately 0.5927 for site percolation on a square lattice), only small finite clusters exist. At p_c, the system undergoes a continuous phase transition: a giant connected cluster suddenly emerges, creating a top-to-bottom spanning path across the lattice.

Key observables include the spanning probability P_inf(p), which jumps from 0 to 1 near p_c; the average cluster size, which diverges at the critical point; and the cluster size distribution n(s), which follows a power law n(s) ~ s^(-tau) at p_c with tau = 187/91 in 2D. This critical exponent tau is universal -- it depends only on the spatial dimension, not on the lattice details.

Percolation models appear throughout science and engineering: forest fire spread (can fire cross the forest?), material conductivity (do conducting paths form in a composite?), oil recovery (can water flood through rock pores?), network resilience (does the internet stay connected when routers fail?), and epidemic thresholds (does disease spread through a population?). The sharp phase transition makes percolation a paradigmatic model of critical phenomena in statistical physics.

Use the occupation probability slider to control how densely the lattice is filled. Watch the dramatic visual change as p crosses the critical threshold near 0.593. The spanning indicator shows whether a connected path exists from top to bottom. Try the preset scenarios: Below Critical (p=0.4) shows isolated clusters, Near Critical (p=0.593) shows the fractal transition, Above Critical (p=0.7) shows a dominant spanning cluster. The Critical Sweep animation automatically sweeps through the transition region for a dramatic visualization of the phase change.