FKG Inequality - Beyond Product Measures

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:

\Phi_\Lambda(\phi)=\begin{cases} \beta 1_{\{\phi(x)\not=\phi(y)\}} & \text{ if }\Lambda=\{x,y\}\text{ is a pair of adjacent vertices of }\Gamma;\\ 0 & \text{ otherwise.}\end{cases}

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).

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