Introduction to Percolation Theory (3) Supercritical and Subcritical States

Proving the Existence of Supercritical and Subcritical States in Percolation

In this chapter, we prove the existence of supercritical and subcritical states in percolation models on -dimensional cubic lattices. Specifically, we will establish that for . To prove this theorem, we break the task into two steps:

  1. Prove that .
  2. Prove that .

By using the property , we can conclude the theorem.

Step 1: Prove that

The general idea for this proof is as follows: If the origin belongs to an infinite open cluster, then we can always find a self-avoiding open path of infinite length starting at the origin. Hence, if the event “the origin belongs to an infinite open cluster” occurs with positive probability, then the event “such a self-avoiding open path exists” also occurs with positive probability. This reduces the problem of studying infinite open clusters to counting the number of such self-avoiding open paths.

Since studying paths of infinite length is difficult, we first examine paths of fixed length and then let . Let denote the number of self-avoiding open paths of length starting from the origin in , and let denote the number of self-avoiding paths of length starting from the origin (regardless of whether the edges are open). Then:

If the origin belongs to an infinite open cluster, there must be at least one such path.

Using a standard trick in probability theory:

Noting that:

(where counts the number of self-avoiding paths of length , and is the probability that all edges along such a path are open), we obtain:

To proceed, we need to understand the growth of as . Our goal is to bound by a constant raised to the power of , enabling us to combine it with the term.

Clearly, is monotonically increasing with . In , starting from the origin , there are adjacent vertices available for the next step in the path. At , there are at most adjacent vertices to choose from (excluding to ensure the path is self-avoiding). By the multiplication principle:

Substituting this into our earlier inequality:

We aim to show that for sufficiently small , . That is, as , approaches . Notice:

Letting , we see that when . This proves:

To further refine this bound, define the connectivity constant for as:

Clearly, as . Substituting this refinement, we improve the bound:

Step 2: Prove that

To prove , we aim to find a constant such that when . This ensures that for sufficiently large , is positive.

We observe that any finite open cluster is finite because all outward edges of the cluster’s boundary vertices are closed. Conversely, an infinite open cluster exists because at least one boundary vertex’s outward edges can extend infinitely. Hence, finite open clusters are always “surrounded” by minimal enclosing closed loops, while infinite open clusters lack such enclosing loops. This leads us to study closed loops instead of infinite open clusters.

We construct the dual graph of . The vertices of the dual graph correspond to the centers of squares in , and an edge in the dual graph is open if and only if it crosses a closed edge in the original graph. Then, the finite open cluster containing the origin in the original graph is surrounded by a closed loop in the dual graph.

Let denote the number of loops in the dual graph of length enclosing the origin. Each loop can be “cut” into a path by removing one edge. Such paths correspond to self-avoiding paths of length in the dual graph. Hence:

Substituting this into the inequality for , we have:

where the summation is over all loops . Estimating :

For large , , so:

This summation converges when . Hence, for , . This proves:

Conclusion

Combining the results:

This establishes the existence of supercritical and subcritical states for . In subsequent chapters, we will introduce useful inequalities in probabilistic graph models as tools for further research.