2 - Percolation

Summary

Random fractals in Nature arise for a variety of reasons (dynamic chaotic processes, self-organized criticality, etc.) that are the focus of much current research. Percolation is one such chief mechanism. The importance of percolation lies in the fact that it models critical phase transitions of rich physical content, yet it may be formulated and understood in terms of very simple geometrical concepts. It is also an extremely versatile model, with applications to such diverse problems as supercooled water, galactic structures, fragmentation, porous materials, and earthquakes.

The percolation transition

Consider a square lattice on which each bond is present with probability p, or absent with probability 1 − p. When p is small there is a dilute population of bonds, and clusters of small numbers of connected bonds predominate. As p increases, the size of the clusters also increases. Eventually, for p large enough there emerges a cluster that spans the lattice from edge to edge (Fig. 2.1). If the lattice is infinite, the inception of the spanning cluster occurs sharply upon crossing a critical threshold of the bond concentration, p = p_c.

The probability that a given bond belongs to the incipient infinite cluster, P∞, undergoes a phase transition: it is zero for p < p_c, and increases continuously as p is made larger than the critical threshold p_c (Fig. 2.2).