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.

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 the random cluster model.

Before the FKG inequality, there was a similar idea called the Harris inequality. It was created by Theodore Edward Harris. Since then, mathematicians have created 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.

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 books, 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

The FKG inequality has a simpler form called the Harris inequality for special setups. This happens when the system can be arranged in a line or made 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

The FKG inequality is used in statistical mechanics. It helps us study systems where measures follow special rules.

One example is the Ising model. This model looks at tiny particles called "spins." These spins can be +1 or -1.

The FKG inequality shows us how these spins behave together. It tells us which patterns are more likely to happen at the same time. This is like how, in a group of friends, if one person likes ice cream, others might like it too. The inequality helps us understand these connections in complicated systems.

A generalization: the Holley inequality

The Holley inequality, made by Richard Holley, is a way to compare average values of special functions on something called a lattice. It says that if some rules are followed, the average of one function under one rule will be bigger than the average under another rule. This helps us understand the FKG inequality, which shows how different events in random systems are connected. The Holley inequality comes from another inequality called the Ahlswede–Daykin inequality.

Weakening the lattice condition: monotonicity

When we study 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 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, 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.