FKG inequality

The Fortuin–Kasteleyn–Ginibre (FKG) inequality is an important idea in mathematics, especially in areas like statistical mechanics and probabilistic combinatorics. It helps us understand how different things in random systems are connected to each other. This inequality was created by three mathematicians: Cees M. Fortuin, Pieter W. Kasteleyn, and Jean Ginibre.

In simple terms, the FKG inequality tells us that in many random situations, events that grow or increase tend to support each other. But if one event increases and another decreases, they tend to work against each other. This idea came from studying something called the random cluster model.

Before the FKG inequality, there was a similar idea called the Harris inequality, which applies to special cases where things happen completely by chance. It was created by Theodore Edward Harris. Since then, mathematicians have created even more ways to understand these connections, like the Holley inequality (1974) and the Ahlswede–Daykin "four functions" theorem (1978). The FKG inequality is also related to the Griffiths inequalities, though they work a little differently.

The inequality

The FKG inequality is a special rule that helps us understand how different things change together in random systems. It was discovered by three mathematicians in 1971.

This rule says that when we look at two things that both tend to increase or both tend to decrease, they often affect each other in a positive way. But if one thing increases while the other decreases, they tend to affect each other in a negative way. This idea is useful in studying random graphs and other chance-based systems.

Variations on terminology

The lattice condition for μ is also called multivariate total positivity, and sometimes the strong FKG condition. In older literature, the term (multiplicative) FKG condition is also used.

The property of μ that increasing functions are positively correlated is also called having positive associations, or the weak FKG condition.

So, the FKG theorem can be rephrased as "the strong FKG condition implies the weak FKG condition".

A special case: the Harris inequality

When dealing with special kinds of setups, the FKG inequality has a simpler form called the Harris inequality. This happens when the system can be arranged in a line or built from many independent parts. In these cases, the Harris inequality tells us that certain events are more likely to happen together.

For example, imagine coloring parts of a honeycomb pattern either black or white by chance. If we look for paths of black spots from one place to another, finding one such path makes it more likely to find another path elsewhere. This shows that some events support each other.

Examples from statistical mechanics

In statistical mechanics, the FKG inequality often applies to systems where measures follow certain rules. One common example is the Ising model, which studies how tiny particles called "spins" interact on a graph. These spins can be either +1 or -1.

When we look at how these spins behave together, we find that some patterns are more likely to happen at the same time. This is similar to how, in a group of friends, if one person likes ice cream, others might also like it too. The FKG inequality helps us understand these connections in complex systems.

A generalization: the Holley inequality

The Holley inequality, created by Richard Holley, is a way to compare the average values of special kinds of functions on a mathematical structure called a lattice. It tells us that if certain rules are followed, the average value of one function with respect to one rule will be bigger than the same function's average with respect to another rule. This idea helps us understand the FKG inequality, which shows how different events in random systems are related to each other. The Holley inequality itself comes from another inequality called the Ahlswede–Daykin inequality.

Weakening the lattice condition: monotonicity

When we look at a special kind of math problem where things are arranged in a grid, we can find a simpler way to check if the FKG inequality works. This inequality helps us understand how different parts of a random system are connected.

If a certain math rule called "monotonicity" is true, then the FKG inequality will also be true. One way to show this uses a process called a Markov chain, which updates the grid step by step using random numbers. Because of the monotonicity, each step depends on these random numbers in a way that keeps things connected positively. This means the final result will also show this positive connection.

There is also a way to compare two different math rules using this idea of monotonicity. If one rule always does at least as well as another in a certain way, we can show this using a similar Markov chain process. This helps prove another important math result called the Holley inequality, which in turn helps prove the FKG inequality.