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:
“The books may say that nine-month-olds crawl, say their first words, and are afraid of strangers. Your exuberantly concrete and special nine-month-old hasn’t read them. She may be walking already, not saying a word and smiling gleefully at every stranger she sees. . . . You can support her best by helping her learn what she’s trying to learn, not what the books say a typical child ought to be learning.”
—Amy Laura Dombro (20th century)
“Mother came to us destitute. She brings a child into the world, takes one look at him and promptly dies. Without leaving so much as a forwarding name and address.”
—Vernon Harris (c. 1910)
“A man willing to work, and unable to find work, is perhaps the saddest sight that fortune’s inequality exhibits under this sun.”
—Thomas Carlyle (1795–1881)