A Special Case: The Harris Inequality
If the lattice is totally ordered, then the lattice condition is satisfied trivially for any measure μ. For this case, the FKG inequality is Chebyshev's sum inequality: if the two increasing functions take on values and, then (we may assume that the measure μ is uniform)
More generally, for any probability measure μ on and increasing functions ƒ and g,
which follows immediately from
The lattice condition is trivially satisfied also when the lattice is the product of totally ordered lattices, and is a product measure. Often all the factors (both the lattices and the measures) are identical, i.e., μ is the probability distribution of i.i.d. random variables.
The FKG inequality for the case of a product measure is known also as the Harris inequality after Harris (Harris 1960), who found and used it in his study of percolation in the plane. A proof of the Harris inequality that uses the above double integral trick on can be found, e.g., in Section 2.2 of Grimmett (1999).
Read more about this topic: FKG Inequality
Famous quotes containing the words special, harris and/or inequality:
“For universal love is as special an aspect as carnal love or any of the other kinds: all forms of mental and spiritual activity must be practiced and encouraged equally if the whole affair is to prosper. There is no cutting corners where the life of the soul is concerned....”
—John Ashbery (b. 1927)
“Mr. Brownlow: The law supposes that your wife acts under your direction.
Bumble: If that’s what the law supposes, sir, then the law’s an ass. And if that’s the eye of the law, sir, then the law’s a bachelor.”
—Vernon Harris (c. 1910)
“Nature is unfair? So much the better, inequality is the only bearable thing, the monotony of equality can only lead us to boredom.”
—Francis Picabia (1878–1953)