This Month's Feature Column

Percolation: Slipping through the Cracks

Introduction

My backyard has an area where the soil is mainly clay and another where it's mainly sand. The day after a hard rain, the sandy region is usually dry while the clay region is still damp. The process by which water moves through a medium, like sand or clay, is called percolation and is currently the focus of significant mathematical activity, some of which we'll describe in this article.

Geoffrey Grimmett begins his book Percolation with the question: "Suppose we immerse a large porous stone in a bucket of water. What is the probability that the centre of the stone is wetted?" To begin creating a mathematical model, we will imagine a two-dimensional lattice of channels running through the rock (a more realistic three-dimensional model can wait).

This is known as the square lattice, and we will denote it by Z². We choose a parameter p between 0 and 1 and declare that each edge is open with probability p. Think of an open edge as a channel that is large enough to conduct water through it. Here are examples for two different values of p.

p = 0.3	p = 0.6

If we imagine that the size of the channels is much smaller than the size of the rock, it is reasonable to assume that the lattice is infinite in extent. To rephrase Grimmett's question, we may ask, "What is the probability that there is a path of open edges from the origin that travels infinitely far?

We can easily make two statements. When p = 0, every edge is closed so there will be no infinite open path containing the origin. However, when p = 1, every edge will be open so there must be an infinite open path.

What happens for intermediate values of p? For small p, there will be few open channels so any open paths will most likely be short. However, as p increases, there are more open channels, and eventually it is likely that there is an infinite open path starting at the origin. If there is a positive probability of having such an infinite path, we say that percolation occurs. We will see that there is a critical probability, that we denote p_c, representing a threshold; percolation occurs above p_c but not below. A result due to Harris and Kesten, which we'll outline later, states that the critical probability for the square lattice is p_c = 1/2.

The existence of a critical probability makes percolation a mathematically interesting and rich subject. On either side of the critical probability, the system behaves in fundamentally different ways (as water drains easily through sandy soil but not through clay). As such, it serves as a model for more complex systems that experience a phase transition when some parameter, such as temperature, passes through a critical value. Percolation provides a model that is simple enough to be mathematically accessible while still displaying many of the features of more complex systems.

David Austin
Grand Valley State University
david at merganser.math.gvsu.edu