Beyond Product Measures
In statistical mechanics, the usual source of measures that satisfy the lattice condition (and hence the FKG inequality) is the following:
If is an ordered set (such as ), and is a finite or infinite graph, then the set of -valued configurations is a poset that is a distributive lattice.
Now, if is a submodular potential (i.e., a family of functions
one for each finite, such that each is submodular), then one defines the corresponding Hamiltonians as
If μ is an extremal Gibbs measure for this Hamiltonian on the set of configurations, then it is easy to show that μ satisfies the lattice condition, see Sheffield (2005).
A key example is the Ising model on a graph . Let, called spins, and . Take the following potential:
Submodularity is easy to check; intuitively, taking the min or the max of two configurations tends to decrease the number of disagreeing spins. Then, depending on the graph and the value of, there could be one or more extremal Gibbs measures, see, e.g., Georgii, Häggström & Maes (2001) and Lyons (2000).
Read more about this topic: FKG Inequality
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