Introduction to Percolation Theory (6) BK Inequality

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2024-08-21

Academics

The BK Inequality for Increasing Events

Today, we introduce another inequality for estimating probabilities of increasing events: the BK inequality. While the FKG inequality intuitively demonstrates the properties and relationships of increasing events, it may not always provide the desired inequality direction for certain applications. The BK inequality addresses this by establishing bounds for the disjoint occurrence of two increasing events and :

where denotes the event that and occur disjointly.

Definition of Disjoint Occurrence

The event is defined as the disjoint occurrence of and . Specifically, given an edge configuration , if the set of open edges can be partitioned into two disjoint subsets such that occurs in one subset and occurs in the other, then occurs.

Formally, let represent the edge configuration, and let denote the set of open edges. If there exists a partition such that:

for with , and
for with ,

then .

Intuition for Disjoint Occurrence

To simplify the mathematical abstraction, consider the following intuitive example. Let be a finite subgraph of the cubic lattice, and define the event as “there exists an open path from to entirely within .” Clearly, is an increasing event.

Now consider , which represents the event “there exists an open path from to and an open path from to , with these two paths being edge-disjoint.” Intuitively:

This is because the edge-disjoint constraint in creates additional restrictions compared to . Simplifying further:

This perfectly aligns with the BK inequality.

Statement of the BK Inequality

For increasing events and that depend on a finite number of edges:

Proof Outline

The proof involves weakening the dependencies between and to approach independence. The core idea is to replace each edge in the cubic lattice with multiple parallel edges. For example, if each original edge is replaced by , with being a sufficiently large constant:

Assign to event , and
Assign to event .

This separation reduces the dependency between and , making them nearly independent. For the modified graph:

Moreover, the disjoint occurrence is not hindered by adding new edges, as additional edges facilitate the disjoint occurrence of and . Hence, the BK inequality holds:

Extensions of the BK Inequality

Associativity of :
The operator is associative, allowing the BK inequality to extend to multiple events. For increasing events that depend on a finite number of edges:

Example: Application in Percolation

Consider , where each is a collection of paths within the percolation model. Define as the event “there exists an open path in .” Clearly, is an increasing event, and each depends only on a finite number of edges.

Now, the event represents the disjoint occurrence of open paths in each , meaning there exist paths such that these paths are pairwise edge-disjoint. On the other hand, represents the event “there exists an open path in each ,” with no restriction on edge disjointness.

Using the BK inequality:

Summary

The BK inequality complements the FKG inequality by providing bounds for the probabilities of disjoint occurrences of increasing events. This additional perspective is particularly useful in scenarios where edge-disjoint constraints are involved. In the next chapter, we will explore more advanced probabilistic tools for studying percolation models.