Percolation theory

Percolation theory is the science of "clumping" in random environments. Suppose a square lattice is randomly filled with black and white sites. A clump or cluster is a spatially connected group of white sites. Two white sites belong to the same cluster if (and only if) it is possible to move from one site to the other by repeatedly stepping to an adjacent white site without ever touching a black site.

Fig.1: A cluster is a group of white sites in which every two sites s, t can be connected by a path that consists entirely of white sites. Metaphorically speaking, if white sites are "land" and black sites "water", a cluster is an island surrounded by an ocean.
(The white site in the upper left corner of the lattice does not belong to the highlighted cluster because only steps in the four principal directions – up, down, left, right – but not in the diagonal directions, are permitted in this example.)

The main interest of percolation theory are the sizes of the clusters. In particular, is there a cluster that is large enough to connect the top to the bottom of the lattice?
Two extreme cases are immediately obvious. If there are very few white sites, there is probably no such path because all clusters are small compared to the entire lattice. On the other hand, if almost all sites are white, there are many possible paths.

Fig.2: (a) If there are only a few white sites, clusters are too small to connect opposite sides of the square lattice. (b) If the number of white sites is sufficiently large, it is possible to connect the top and the bottom of the lattice, for example with the highlighted path.

What happens between these extremes?
Let us call the probability that a site is white p. Otherwise the site is black with probability 1-p.
The probability P_path that there is a path from top to bottom depends on p and the linear size L of the lattice (i.e., the lattice consists of L×L little squares). As L increases, P_path approaches a very intriguing limiting behaviour. Below a critical value p_c ≈ 0.593, P_path is approximately zero. On the other hand, for all p larger than p_c, P_path is almost equal to 1. This means that for large lattices there is a sudden change in the global connectivity: no paths for p < p_c, but almost certainly at least one path for p > p_c.

Fig.3: The probability P_path that there is a path between opposite sites of a L×L square lattice.

In the limit of infinite L, P_path becomes a step function, jumping from 0 to 1 at p_c. Such a situation where one goes from the impossible (P_path=0) to the inevitable (P_path=1), without ever visiting the improbable, is called a "0-1 law" in mathematics. In physics, this phenomenon is called a "phase transition."

Many variations of the problem sketched above are possible. For example, the square lattice can be substituted by a triangular, hexagonal or even more complicated lattice. The problem can be extended to three or more dimensions. Or, instead of the sites, we could colour the bonds between adjacent sites white or black.

Fig.4a: triangular lattice

Fig.4b: three-dimensional cubic lattice

Fig.4c: bond percolation on a square lattice

I am interested in the extension of percolation theory in two directions.

In the standard treatment of percolation, p is constant for every site in the lattice. For many applications, it is more natural to allow p to vary across space.
The states of adjacent sites are usually considered to be independent: the fact that one site is white has no consequence for the probability that its neighbours are also white. However, in realistic cases, the states are often correlated.

Selected publications:

Michael T. Gastner, Beata Oborny, Alexey B. Ryabov, and Bernd Blasius
Changes in the gradient percolation transition caused by an Allee effect
Physical Review Letters 106, 128103 (2011). (Preprint available)

Michael T. Gastner, Beata Oborny, D. K. Zimmermann, and Gunnar Pruessner
Transition from connected to fragmented vegetation across an environmental gradient: scaling laws in ecotone geometry
The American Naturalist, 174, E23-E39 (2009). (available as free E-Article)