6 - Percolation, spanning trees and the Potts model

Summary

In this chapter we make a side step to a subject which is not directly concerned with polymers but which is closely connected to it. It is the study of percolation. Together with problems such as SAWs and lattice animals percolation forms a subject which is sometimes called geometrical critical phenomena. As we will see in chapter 8, percolation (in d = 2) is closely related to the behaviour of polymers at the ‘θ-point’ Moreover, percolation plays a role in the collapse of branched polymers. We will also discuss how percolation is related to a spin model (the Potts model) just as polymers are related to the O(n)-model. This Potts model turns out to be of importance in the description of dense polymer systems (chapter 7). The Potts model can also be used to describe the so called spanning trees. These are in turn interesting in the study of branched polymers (chapter 9).

In this book we have to limit ourselves to a discussion of those properties of percolation and Potts models which are necessary for the sequel of this book. There exists excellent reviews and books about percolation and for more information we refer to these.

Percolation as a critical phenomenon

To introduce percolation, think of a regular lattice, e.g. the hypercubic lattice in d dimensions. Take a real number 0 ≤ p ≤ 1 which we call the occupation probability. We occupy either the vertices (sites) or the edges (bonds) of this lattice with probability p.