FKG Inequality - A Special Case: The Harris Inequality

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:

    A special kind of beauty exists which is born in language, of language, and for language.
    Gaston Bachelard (1884–1962)

    The deadly monotony of Christian country life where there are no beggars to feed, no drunkards to credit, which are among the moral duties of Christians in cities, leads as naturally to the outvent of what Methodists call “revivals” as did the backslidings of the people in those days.
    —Corra May Harris (1869–1935)

    All the aspects of this desert are beautiful, whether you behold it in fair weather or foul, or when the sun is just breaking out after a storm, and shining on its moist surface in the distance, it is so white, and pure, and level, and each slight inequality and track is so distinctly revealed; and when your eyes slide off this, they fall on the ocean.
    Henry David Thoreau (1817–1862)