7 Gibbs states Brook’s theorem states that a positive probability measure on a finite product may be decomposed into factors indexed by the cliquesofits dependency graph. Closely related to this is the well known fact that apositivemeasureisaspatialMarkovfieldonagraphG if and only if it is a Gibbs state. The Ising and Potts models are introduced, and the n-vector model is mentioned. 7.1 Dependency graphs Let X (X1, X2,...,Xn) be a family of random variables on a given probability= space. For i, j V 1, 2,...,n with i j,wewritei j ∈ ={ } "= ⊥ if: X and X are independent conditional on (X : k i, j).Therelation i j k "= is thus symmetric, and it gives rise to a graph G with vertex set V and edge-set⊥ E i, j : i j ,calledthedependency graph of X (or of its ={$ % "⊥ } law). We shall see that the law of X may be expressed as a product over terms corresponding to complete subgraphs of G.Acompletesubgraphof G is called a clique,andwewriteK for the set of all cliques of G.For notational simplicity later, we designate the empty subset of V to be a clique, and thus ∅ K.Acliqueismaximal if no strict superset is a clique, and ∈ we write M for the set of maximal cliques of G. Weassumeforsimplicitythatthe Xi take values in some countable subset S of the reals R.ThelawofX gives rise to a probability mass function π on Sn given by π(x) P(X x for i V ), x (x , x ,...,x ) Sn. = i = i ∈ = 1 2 n ∈ It is easily seen by the definition of independence that i j if and only if π may be factorized in the form ⊥ π(x) g(x , U)h(x , U), x Sn, = i j ∈ for some functions g and h,whereU (x : k i, j).ForK K and = k "= ∈ x Sn,wewritex (x : i K ).Wecallπ positive if π(x)>0forall ∈ K = i ∈ x Sn. ∈ K In the following, each function fK acts on the domain S . 142 7.1 Dependency graphs 143 7.1 Theorem [54]. Let π be a positive probability mass function on Sn. There exist functions f : S K [0, ),K M,suchthat K → ∞ ∈ (7.2) π(x) f (x ), x Sn. = K K ∈ K M !∈ In the simplest non-trivial example, let us assume that i j whenever i j 2. The maximal cliques are the pairs i, i 1 ,andthemass⊥ | − |≥ { + } function π may be expressed in the form n 1 − n π(x) fi (xi , xi 1), x S , = + ∈ i 1 != so that X is a Markov chain, whatever the direction of time. Proof. We shall show that π may be expressed in the form (7.3) π(x) f (x ), x Sn, = K K ∈ K K !∈ for suitable fK .Representation(7.2)followsfrom(7.3)byassociatingeach fK with some maximal clique K * containing K as a subset. Arepresentationofπ in the form π(x) f (x) = r r ! is said to separate i and j if every fr is a constant function of either xi or xj ,thatis,no fr depends non-trivially on both xi and xj .Let (7.4) π(x) f (x ) = A A A A !∈ be a factorization of π for some family A of subsets of V ,andsupposethat i, j satisfies: i j,buti and j are not separated in (7.4). We shall construct ⊥ from (7.4) a factorization that separates every pair r, s that is separated in (7.4), and in addition separates i, j.Continuingbyiteration,weobtaina factorization that separates every pair i, j satisfying i j,andthishasthe required form (7.3). ⊥ Since i j, π may be expressed in the form ⊥ (7.5) π(x) g(x , U)h(x , U) = i j for some g, h,whereU (x : j i, j).Fixs, t S,andwriteh = k "= ∈ t (respectively, h )forthefunctionh(x) evaluated with x t (respectively, s,t j = " x s, x t). By (7.4), " i = j = " " π(x) π(x) (7.6) π(x) π(x) f A(xA) . = t π(x) = t π(x) t #A A $ t " !∈ " " " " " " " 144 Gibbs states By (7.5), the ratio π(x) h(xj , U) = ( , ) π(x) t h t U is independent of xi ,sothat " " π(x) f A(xA) s . π(x) = f (x ) t A A A A "s,t !∈ " By (7.6), " " " " f A(xA) s π(x) f A(xA) = t f (x ) #A A $#A A A A "s,t $ !∈ " !∈ " is the required representation, and the" claim is proved." ! " 7.2 Markov and Gibbs random fields Let G (V, E) be a finite graph, taken for simplicity without loops or multiple= edges. Within statistics and statistical mechanics, there has been agreatdealofinterestinprobabilitymeasureshavingatypeof‘spatial Markov property’ given in terms of the neighbour relation of G.Weshall restrict ourselves here to measures on the sample space " 0, 1 V ,while ={ } noting that the following results may be extended without material difficulty to a larger product SV ,whereS is finite or countably infinite. The vector σ " may be placed in one–one correspondence with the ∈ subset η(σ) v V : σv 1 of V ,andweshallusethiscorrespondence ={ ∈ = } freely. For any W V ,wedefinetheexternal boundary ⊆ %W v V : v/W,v w for some w W . ={ ∈ ∈ ∼ ∈ } For s (sv : v V ) ",wewritesW for the sub-vector (sw : w W ). We refer= to the configuration∈ ∈ of vertices in W as the ‘state’ of W . ∈ 7.7 Definition. Aprobabilitymeasureπ on " is said to be positive if π(σ) > 0forallσ ".ItiscalledaMarkov (random) field if it is positive and: for all W V∈,conditionalonthestateofV W ,thelawofthestateof ⊆ \ W depends only on the state of %W .Thatis,π satisfies the global Markov property (7.8) π σW sW σV W sV W π σW sW σ%W s%W , = \ = \ = = = for all s %",andW " V . & % " & ∈ ⊆" " The key result about such measures is their representation intermsofa ‘potential function’ φ,inaformknownasaGibbsrandomfield(orsome- times ‘Gibbs state’). Recall the set K of cliques of the graph G,andwrite 2V for the set of all subsets (or ‘power set’) of V . 7.2 Markov and Gibbs random fields 145 7.9 Definition. Aprobabilitymeasureπ on " is called a Gibbs (random) V field if there exists a ‘potential’ function φ :2 R,satisfyingφC 0if C / K,suchthat → = ∈ (7.10) π(B) exp φ , B V. = K ⊆ #K B $ '⊆ We allow the empty set in the abovesummation, so that log π(∅) φ∅. = Condition (7.10) has been chosen for combinatorial simplicity. It is not the physicists’ preferred definition of a Gibbs state. Letusdefinea Gibbs state as a probability measure π on " such that there exist functions f : 0, 1 K R, K K,with K { } → ∈ (7.11) π(σ) exp f (σ ) ,σ". = K K ∈ #K K $ '∈ It is immediate that π satisfies (7.10) for some φ whenever it satisfies (7.11). The converse holds also, and is left for Exercise 7.1. Gibbs fields are thus named after Josiah Willard Gibbs, whose volume [95] made available the foundations of statistical mechanics. A simplistic motivation for the form of (7.10) is as follows. Suppose that each state σ has an energy Eσ ,andaprobabilityπ(σ).Weconstraintheaverageenergy E Eσ π(σ) to be fixed, and we maximize the entropy = σ ( η(π) π(σ)log π(σ). =− 2 σ " '∈ With the aid of a Lagrange multiplier β,wefindthat β Eσ π(σ) e− ,σ". ∝ ∈ The theory of thermodynamics leads to the expression β 1/(kT) where k is Boltzmann’s constant and T is (absolute) temperature.= Formula (7.10) arises when the energy Eσ may be expressed as the sum of the energies of the sub-systems indexed by cliques. 7.12 Theorem. Apositiveprobabilitymeasureπ on " is a Markov random field if and only if it is a Gibbs random field. The potential function φ corresponding to the Markov field π is given by K L φ ( 1)| \ | log π(L), K K. K = − ∈ L K '⊆ Apositiveprobabilitymeasureπ is said to have the local Markov property if it satisfies the global property (7.8) for all singleton sets W and all s ". ∈ The global property evidently implies the local property, and it turns out that the two properties are equivalent. For notational convenience, we denote a singleton set w as w. { } 146 Gibbs states 7.13 Proposition. Let π be a positive probability measure on ".The following three statements are equivalent: (a) π satisfies the global Markov property, (b) π satisfies the local Markov property, (c) for all A Vandanypairu,v Vwithu/ A, v Aandu" v, ⊆ ∈ ∈ ∈ π(A u) π(A u v) (7.14) ∪ ∪ \ . π(A) = π(A v) \ Proof. First, assume (a), so that (b) holds trivially. Let u / A, v A,and ∈ ∈ u " v.Applying(7.8)withW u and, for w u, sw 1ifandonlyif ={ } "= = w A,wefindthat ∈ (7.15) π(A u) ∪ π(σu 1 σV u A) π(A) π(A u) = = | \ = + ∪ π(σ 1 σ% A %u) = u = | u = ∩ π(σu 1 σV u A v) since v/%u = = | \ = \ ∈ π(A u v) ∪ \ . = π(A v) π(A u v) \ + ∪ \ Equation (7.15) is equivalent to (7.14), whence (b) and (c) are equivalent under (a). It remains to show that the local property implies the global property. The proof requires a short calculation, and may be done eitherbyTheorem 7.1 or within the proof of Theorem 7.12. We follow the first route here. Assume that π is positive and satisfies the local Markov property. Then u v for all u,v V with u " v.ByTheorem7.1,thereexistfunctions f ⊥, K M,suchthat∈ K ∈ (7.16) π(A) f (A K ), A V.