An ?Inner? Variational Principle for Markov Fields on a Graph

Abstract

Consider an infinite graph A and the set f2=S A of all collective states co: A--*S which specify for each site a~A an individual state co(a) in some finite state space S. A Markov field on A is a probability measure on f2 such that for each asA the conditional probability rc~(s ] q) of co(a) = s, given the states t/(b) of all the other sites b + a, actually depends only on the states of the nearest neighbors of a. Let us call the collection of kernels H = (~)a~A the local characteristics of the Markov field. It is well known that in general a Markov field is not determined by its local characteristics, and one says that/7 admits a phase transition if there are at least two Markov fields with local characteristics/ 7; cf. [3]. IfA is the d-dimensional lattice Z d then the translation invariant Markov fields on A with fixed local characteristics H are characterized by the fact that they minimize the functional f = e- h, where h(p) denotes the specific entropy of # and where e(/~), the specific energy of#, is defined in terms of//; cf. [2]. This can be translated into information theoretical terms: two translation invariant Markov fields # and v have the same local characteristics if and only if the specific information gain h(#[v) of/~ with respect to v is equal to 0; cf. [1]. There are, however, homogeneous graphs other than Z d where one can introduce the thermodynamical quantities h(#), e(#) and h(#] v), but where the variational principle above breaks down. Consider, for example, the d-dimensional Cayley tree A=T a with d+l branches emanating from every vertex. For d>2 there are different positive transition matrices P(., .) and Q(., .) on S= {0, 1} such that the associated Markov chain # and v on ~2 = S are Markov fields with the same local characteristics/ 7 = (rCa)a~A; cf. [3, 5]. On the other hand it is easy to see that for two such measures the specific information gain is given by h(#l v) = ~ #[co(a) = s] H(P(s, .)l Q(s, .)) s~S