Short-range and long-range order: a transition in block-gluing behavior in Hom shifts

Abstract Hom shifts form a class of multidimensional shifts of finite type (SFT) and consist of colorings of the grid ℤ 2 where adjacent colors must be neighbors in a fixed finite undirected simple graph G . This class includes several important statistical physics models such as the hard square model. The gluing gap measures how far any two square patterns of size n can be glued, and it can be seen as a measure of the range of order which affects the possibility to compute the entropy (or free energy per site) of a shift. This motivates a study of the possible behaviors of the gluing gap. The class of Hom shifts is interesting because it allows us to describe dynamical properties, in particular mixing-type ones in the context of this article, using algebraic graph theory, which has received a lot of attention recently. Improving some former work of N. Chandgotia and B. Marcus, we prove that the gluing gap either depends linearly on n or is dominated by log( n ). We also find a Hom shift with gap Θ(log( n )), confirming a conjecture formulated by R. Pavlov and M. Schraudner. The physical interest of these results is to better understand the transition from short-range to long-range order (respectively sublogarithmic and linear gluing gap), which is reflected in whether some parameter, the square cover, is finite or infinite.