Book contents
- Frontmatter
- Contents
- Preface
- 1 Random walks on graphs
- 2 Uniform spanning tree
- 3 Percolation and self-avoiding walk
- 4 Association and influence
- 5 Further percolation
- 6 Contact process
- 7 Gibbs states
- 8 Random-cluster model
- 9 Quantum Ising model
- 10 Interacting particle systems
- 11 Random graphs
- 12 Lorentz gas
- References
- Index
5 - Further percolation
Published online by Cambridge University Press: 05 June 2012
- Frontmatter
- Contents
- Preface
- 1 Random walks on graphs
- 2 Uniform spanning tree
- 3 Percolation and self-avoiding walk
- 4 Association and influence
- 5 Further percolation
- 6 Contact process
- 7 Gibbs states
- 8 Random-cluster model
- 9 Quantum Ising model
- 10 Interacting particle systems
- 11 Random graphs
- 12 Lorentz gas
- References
- Index
Summary
The subcritical and supercritical phases of percolation are characterized respectively by the absence and presence of an infinite open cluster. Connection probabilities decay exponentially when p < pc, and there is a unique infinite cluster when p > pc. There is a power-law singularity at the point of phase transition. It is shown that pc = ½ for bond percolation on the square lattice. The Russo–Seymour–Welsh (RSW) method is described for site percolation on the triangular lattice, and this leads to a statement and proof of Cardy's formula.
Subcritical phase
In language borrowed from the theory of branching processes, a percolation process is termed subcritical if p < pc, and supercritical if p > pc.
In the subcritical phase, all open clusters are (almost surely) finite. The chance of a long-range connection is small, and it approaches zero as the distance between the endpoints diverges. The process is considered to be ‘disordered’, and the probabilities of long-range connectivities tend to zero exponentially in the distance. Exponential decay may be proved by elementary means for sufficiently small p, as in the proof of Theorem 3.2, for example. It is quite another matter to prove exponential decay for all p < pc, and this was achieved for percolation by Aizenman and Barsky and Menshikov around 1986.
- Type
- Chapter
- Information
- Probability on GraphsRandom Processes on Graphs and Lattices, pp. 81 - 126Publisher: Cambridge University PressPrint publication year: 2010