Random Fields: Textures and Classical Statistical Mechanics of Spin Systems

Abstract

If one associates a random variable with each point of a lattice in a one-, two-, or, in general, n-dimensional space the collection of these random variables is called a random field. The interdependence among the random variables of such a field will be significant in determining the properties of macroscopic quantities of such random fields. The interdependence will be restricted in Sect. 4.1 in such a way that we can formulate relatively simple models known as Markov fields. A second approach to simple models for random fields, consisting in the introduction of Gibbs fields, will prove to be equivalent. In Sect. 4.2, examples of Markov random fields will be presented. It will turn out that all classical spin models relevant in statistical mechanics are Markov random fields. Moreover, many two- or three-dimensional images, so-called textures, may also be interpreted as realizations of such fields.