Introduction to Percolation Theory (5) FKG Inequality

The FKG Inequality for Increasing Events

Today, we discuss the FKG inequality related to increasing events. As mentioned in the previous section, increasing events imply that the connectivity of the percolation model and the likelihood of the event occurring change in the same direction. Conversely, for decreasing events, the connectivity of the percolation model and the likelihood of the event occurring change in opposite directions.

Using this concept, we can estimate the relationship between two monotonic events (a term encompassing both increasing and decreasing events) with the FKG inequality.

The FKG Inequality

If and are both increasing events, then an improvement in the connectivity of the percolation model increases both and . This means that these probabilities are positively correlated. Formally:

Using the definition of conditional probability, this leads to:

This is the FKG inequality. The intuition and conclusion are remarkably straightforward.

Similarly, for increasing random variables and satisfying and , their values also exhibit positive correlation:

Equivalently:

Proof of the FKG Inequality

We present the proof for monotonic random variables. For monotonic events, consider the indicator function of the event, and the result directly follows.

Step 1: Finite Dependencies

To simplify the proof, we begin with the case where and depend only on the states of a finite set of edges . We prove the result by induction on .

  • Base Case ():
    Here, and depend only on the state of a single edge . Suppose and are functions of . The edge state takes values with probability and with probability . Thus:


    Therefore:

    Since and are increasing, the above expression is non-negative.

  • Inductive Step:
    Assume the inequality holds for all and depending on edges. For edges, let and depend on . Using the law of total expectation:

    Conditioned on , the state of remains variable, and and are still increasing random variables. By the induction hypothesis:

    Taking the expectation again:

    Using the law of total expectation:

    The induction is complete.

Step 2: Infinite Dependencies

For the infinite case, let and almost surely in , where and depend only on a finite set of edges. Assume:
$$
E_p[X_n] \to E_p[X], \quad E_p[Y_n] \to E_p[Y], \quad