Weakening The Lattice Condition: Monotonicity
Consider the usual case of being a product for some finite set . The lattice condition on μ is easily seen to imply the following monotonicity, which has the virtue that it is often easier to check than the lattice condition:
Whenever one fixes a vertex and two configurations φ and ψ outside v such that for all, the μ-conditional distribution of φ(v) given stochastically dominates the μ-conditional distribution of ψ(v) given .
Now, if μ satisfies this monotonicity property, that is already enough for the FKG inequality (positive associations) to hold.
Here is a rough sketch of the proof, due to Holley (1974): starting from any initial configuration on, one can run a simple Markov chain (the Metropolis algorithm) that uses independent Uniform random variables to update the configuration in each step, such that the chain has a unique stationary measure, the given μ. The monotonicity of μ implies that the configuration at each step is a monotone function of independent variables, hence the product measure version of Harris implies that it has positive associations. Therefore, the limiting stationary measure μ also has this property.
The monotonicity property has a natural version for two measures, saying that μ1 conditionally pointwise dominates μ2. It is again easy to see that if μ1 and μ2 satisfy the lattice-type condition of the Holley inequality, then μ1 conditionally pointwise dominates μ2. On the other hand, a Markov chain coupling argument similar to the above, but now without invoking the Harris inequality, shows that conditional pointwise domination, in fact, implies stochastically domination. Stochastic domination is equivalent to saying that for all increasing ƒ, thus we get a proof of the Holley inequality. (And thus also a proof of the FKG inequality, without using the Harris inequality.)
See Holley (1974) and Georgii, Häggström & Maes (2001) for details.
Read more about this topic: FKG Inequality
Famous quotes containing the word weakening:
“What is wantedwhether this is admitted or notis nothing less than a fundamental remolding, indeed weakening and abolition of the individual: one never tires of enumerating and indicting all that is evil and inimical, prodigal, costly, extravagant in the form individual existence has assumed hitherto, one hopes to manage more cheaply, more safely, more equitably, more uniformly if there exist only large bodies and their members.”
—Friedrich Nietzsche (18441900)