Abstract. The well-known Hammersley-Clifford Theorem states (under certain conditions) that any Markov random field is a Gibbs state for a nearest neighbor interaction. In this paper we study Markov random fields for which the proof of the Hammersley-Clifford Theorem does not apply. Following Petersen and Schmidt we utilize the formalism of cocycles for the homoclinic equivalence relation and introduce "Markov cocycles", reparametrizations of Markov specifications. The main part of this paper exploits this to deduce the conclusion of the Hammersley-Clifford Theorem for a family of Markov random fields which are outside the theorem's purview where the underlying graph is Z d . This family includes all Markov random fields whose support is the d-dimensional "3-colored chessboard". On the other extreme, we construct a family of shift-invariant Markov random fields which are not given by any finite range shift-invariant interaction.

Graph homomorphisms from the Z d lattice to Z are functions on Z d whose gradients equal one in absolute value. These functions are the height functions corresponding to proper 3-colorings of Z d and, in two dimensions, corresponding to the 6-vertex model (square ice). We consider the uniform model, obtained by sampling uniformly such a graph homomorphism subject to boundary conditions. Our main result is that the model delocalizes in two dimensions, having no translation-invariant Gibbs measures. Additional results are obtained in higher dimensions and include the fact that every Gibbs measure which is ergodic under even translations is extremal and that these Gibbs measures are stochastically ordered.

Let H be a finite connected undirected graph and H 2 walk be the graph of bi-infinite walks on H; two such walks {xi} i∈Z and {yi} i∈Z are said to be adjacent if xi is adjacent to yi for all i ∈ Z. We consider the question: Given a graph H when is the diameter (with respect to the graph metric) of H 2 walk finite? Such questions arise while studying mixing properties of hom-shifts (shift spaces which arise as the space of graph homomorphisms from the Cayley graph of Z d with respect to the standard generators to H) and are the subject of this paper.

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Abstract. Fix d ≥ 2. Given a finite undirected graph H without self-loops and multiple edges, consider the corresponding 'vertex' shift, Hom(Z d , H) denoted by XH. In this paper we focus on H which is 'four-cycle free'. The two main results of this paper are: XH has the pivot property, meaning that for all distinct configurations x, y ∈ XH which differ only at finitely many sites there is a sequence of configurations x = x 1 , x 2 , . . . , x n = y ∈ XH for which the successive configurations x i , x i+1 differ exactly at a single site. Further if H is connected then XH is entropy minimal, meaning that every shift space strictly contained in XH has strictly smaller entropy. The proofs of these seemingly disparate statements are related by the use of the 'lifts' of the configurations in XH to the universal cover of H and the introduction of 'height functions' in this context. IntroductionBy H we will always denote an undirected graph without multiple edges (and by abuse of notation also denote its set of vertices). In this paper we focus on H which is four-cycle free, that is, it is finite, it has no self-loops and the four-cycle, denoted by C 4 is not a subgraph of H. Fix d ≥ 2. The basic object of study is X H , the space of graph homomorphisms from Z d to H. Here by Z d we will mean both the group and its standard Cayley graph.Such a space of configurations X H , referred to as a hom-shift, can be obtained by forbidding certain patterns on edges of H Z d . If H is a finite graph then X H is a nearest neighbour shift of finite type. In addition it is also 'isotropic' and 'symmetric', that is, given vertices a, b ∈ H if a is not allowed to sit next to b in X H for some coordinate direction, then a is not allowed to sit next to b in all coordinate directions. Most of the concepts related to shift spaces are introduced in Section 2.Related to a shift space X is its topological entropy denoted by h top (X) which measures the growth rate of the number of patterns allowed in X with the size of the underlying shape (usually rectangular). For a given shift space, its computation is a very difficult task (look for instance in [23] and the references within). We will focus on a different aspect: as in [8], a shift space X is called entropy minimal if for all shift spaces Y X, h top (Y ) < h top (X). Thus if a shift space has zero entropy and is entropy minimal then it is a topologically minimal system. For d = 1, it is well known that all irreducible nearest neighbour shifts of finite type are entropy minimal [15]. However not much is known about it in higher dimensions: Shift spaces with a strong mixing property called uniform filling are entropy minimal [32], however nearest neighbour shifts of finite type with weaker mixing properties like block-gluing may not be entropy minimal [2]. If H is a four-cycle free graph then X H is not even block-gluing (we do not prove this but is implied by our results). There has been some recent work [33] which describes some conditions which are equivalent to entropy minimality for s...