Introduction to Percolation Theory (4) Identities of the Probabilistic Graph Model

Important Results in Probabilistic Graph Models

Today, we begin introducing some important results commonly used in probabilistic graph models for subsequent research. We start by defining the concept of increasing events. The motivation behind this definition is to study how changes in the edge-opening probability affect the global properties of the cubic lattice.

When defining , you might have wondered: Is a monotonic function of ? The intuitive answer is yes—when increases, the connectivity of the model improves, and thus the probability that the origin belongs to an infinite open cluster should not decrease. Formalizing this intuition leads us to the definition of increasing events.

Definition: Increasing and Decreasing Events

For an event , if and imply , then is an increasing event. Similarly, by reversing the partial order on edge configurations, we can define decreasing events.

Edge configurations record the open/closed state of every edge in a specific realization of the percolation model. An increasing event implies that the better the connectivity in the percolation model, the more likely the event will occur, while a decreasing event implies the opposite. (For simplicity, most of the following discussions will focus on increasing events.)

Definition: Increasing Random Variables

A random variable is increasing if , . Similarly, the better the connectivity in the percolation model, the larger the value of an increasing random variable.

With the definitions of increasing events and random variables, we can begin to study how changes in the edge-opening probability affect the global properties of the model. Notably, increasing events and increasing random variables act as a “bridge” between the two.

Key Result: Monotonicity of Expectations and Probabilities

Consider an increasing random variable and a simulation process in the percolation model. Let be a vector where each component , generating the random edge configuration . If , then for the fixed vector , it follows that . Consequently, . Taking expectations, and using the monotonicity of expectations, we obtain:

This is an important result! It shows that when increases in a percolation model, the expectation of an increasing random variable must also increase (though not necessarily strictly).

Extending this result to events, let (the indicator function for an event ), and suppose is an increasing event. Then:

This directly implies:

This aligns well with intuition: When increases in a percolation model, the probability of an increasing event must also increase (though not necessarily strictly).

Note: Both results can be extended to the case of decreasing events.

Monotonicity of

As a direct consequence of the above, we have naturally proven that is an increasing function of , because the event “the origin belongs to an infinite open cluster” is an increasing event.

In addition, we can construct many other increasing events and increasing random variables. For example, let represent the event that “there exists an open path in the cubic lattice connecting two fixed points and .” Clearly, this is also an increasing event.

Significance of Increasing Events

In percolation models, the complexity arises from the combination of the cubic lattice structure and the probabilistic nature of edge states. Studying the probabilities of general events can become exceedingly difficult. In this context, increasing/decreasing events are the simplest class of percolation model events. Furthermore, increasing events link percolation models with different edge-opening probabilities , providing a tool to study how the global properties of the model change with .

Due to their fundamental importance, in the following chapters, we will focus on increasing events and introduce useful methods for estimating the probabilities of such events.