Percolation
Pour water on a porous stone. Will it seep through to the bottom? This is the fundamental question of percolation theory. We assign each site on a grid to be open with probability $p$. As we increase $p$, small disconnected clusters form, merge, and suddenly—at a critical probability $p_c$—a macroscopic cluster spans the entire system.
Phase Transition & Criticality
Site percolation on a 2D square lattice exhibits a continuous phase transition. Let $\theta(p)$ be the probability that a given site belongs to the infinite spanning cluster. For $p < p_c$, the system is in a subcritical phase where all clusters are finite, and $\theta(p) = 0$. For $p > p_c$, a unique infinite cluster emerges, and $\theta(p) > 0$.
Near the critical threshold $p_c \approx 0.592746$, the system exhibits scale invariance and universal power-law scaling. The order parameter behaves as:
Where the critical exponent $\beta = 5/36$ is exact for the 2D percolation universality class, independent of the microscopic lattice details.