In percolation models, a fascinating phenomenon called critical phenomena arises. This refers to the existence of a critical probability (where represents the dimension of the cubic lattice ) such that when the edge-opening probability exceeds or falls below this critical probability, the global properties of the graph undergo a fundamental change. To better understand the motivation for defining the critical probability, we first introduce some necessary definitions and theorems:
Definition: Tail -algebra
Let be events. The -algebra is defined as the -algebra generated by under any finite combination of union, intersection, and complement operations. If we define:
$$ \mathbb{F}i = \sigma(A_i, A{i+1}, …, A_n) $$
then the tail -algebra of the sequence is:
Kolmogorov’s 0-1 Law
If is a sequence of independent events, then the tail -algebra generated by this sequence satisfies the following: If an event , then is either or . The proof of this theorem is straightforward (as it ultimately shows that is independent of itself) and will not be detailed here.
Using Kolmogorov’s 0-1 law, we can study the probability of the existence of an infinite open cluster in percolation. Define the event as “the -th edge is open.” Clearly, no finite collection of can determine the event of the existence of an infinite open cluster, as making a finite number of edges closed in an otherwise fully open cubic lattice does not affect the existence of infinite open clusters. This implies that the event “infinite open cluster exists” is in , meaning:
This is an important conclusion!
Definition of Critical Probability
Can we define a critical probability such that:
When , , and
When , ?
To proceed, let us examine the relationship between and . Recall:
If , then , so .
If , then , so .
This tells us that the largest value of for which can be used to define the critical probability. Thus, we define the critical probability as:
From this definition, we see that:
When , , and
When , .
The critical probability completely separates the graph into two distinct states. When , the graph is in a supercritical state, where an infinite open cluster almost surely exists. When , the graph is in a subcritical state, where an infinite open cluster almost surely does not exist. At , the graph is in a critical state, and the existence of an infinite open cluster cannot be universally determined.
Example: Critical Probability in 1 Dimension
Let us first solve a simple question: what is ? Using the Borel-Cantelli Lemma, we can provide an elegant answer.
In one dimension, when , . However, when , due to the independence of the edge states, let be the event that the -th edge to the left of the origin is closed. Then:
By the Borel-Cantelli Lemma, , which means that infinitely many edges to the left of the origin will almost surely be closed. Similarly, infinitely many edges to the right of the origin will also almost surely be closed. This implies that for any chosen vertex, there will almost surely be infinitely many closed edges on both sides, and thus no infinite open cluster can exist. Therefore, we conclude:
This is a trivial result.
General Critical Probability and Dimensionality
For general , more sophisticated techniques are needed for estimation. However, we can establish a simple inequality:
This follows because any realization of percolation on can be embedded in . If the origin belongs to an infinite open cluster in , it must also belong to an infinite open cluster in its embedding within . This ensures that the function for is at least as large as that for at every point. By the definition of critical probability, .
Further Exploration
In the next discussion, we will introduce an important theorem proving that for , . While this avoids the trivial cases of or , it ensures the existence of both supercritical and subcritical states in higher-dimensional percolation models, providing guidance for further research.