FKG inequality

The FKG inequality gives positive association under a log-supermodular probability measure on a finite distributive lattice.

Increasing events under a product probability measure are positively associated:

P (A \cap B) \geq P (A) P (B)

Harris' inequality

Harris' inequality states that two increasing events under an independent Bernoulli product measure are positively correlated.

An event in a partially ordered configuration space is increasing when changing coordinates upward cannot make a configuration leave the event.

There are no discussions about this article yet.

The FKG inequality, named after its contributors Fortuin, Kasteleyn, and Ginibre, is a result in probability theory that provides a relationship among joint distributions of certain random variables, particularly in the context of lattice structures, such as spins in statistical mechanics. It is most commonly applied in the study of lattice models in statistical physics, including the Ising model.