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

TL;DR

The paper studies two-dimensional Hom shifts $X_G$ determined by a finite graph $G$ and analyzes their block-gluing behavior through the gap function $\\gamma_G(n)$. It proves a sharp dichotomy: for every finite connected $G$, the gap is either $\Theta(n)$ or $O(\log n)$, and provides a concrete $\Theta(\log n)$ example via the Ken-katabami graph $K$, thereby answering questions about intermediate growth and disconfirming a prior conjecture. The authors develop a framework based on the universal cover $\mathcal{U}_G$ and its quotient by squares, introducing the square cover $\mathcal{U}_G^{\square}$ to connect algebraic-topological properties of $G$ to dynamical/gluing properties of $X_G$. They further introduce cactus representations to decompose cycles into manageable units, and prove: (i) if $|\mathcal{U}_G^{\square}|=\infty$ then $\gamma_G(n)=\Theta(n)$, and (ii) if $|\mathcal{U}_G^{\square}|<\infty$ then $\gamma_G(n)=O(\log n)$. The results illuminate a rigid three-class landscape (constant, logarithmic, linear) for block-gluing in Hom shifts and link these classes to whether the square cover is finite, with implications for entropy computability and mixing properties in two dimensions.

Abstract

Hom shifts form a class of multidimensional shifts of finite type (SFT) and consist of colorings of the grid Z2 where adjacent colours 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, which can be seen as a measure of the range of order, and 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 has the interest that mixing type properties can be formulated in terms of 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)), infirming 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.

Figures (14)

• Figure 1: Illustration of the proof of Lemma \ref{['lem:localglobal']}: two possible paths from $(i+1,j)$ to $(i,j+1)$ in $\mathbb{U}$. Only the one on the right can be contained in $\mathbb{U}$.
• Figure 2: A possible choice for $v_{l+1}$ and the corresponding $w_l$.
• Figure 3: Illustration of the definition of $\mathbb{V}$ in two different situations.
• Figure 4: On the left: a graph $G$ containing a cycle of length four ($aabca$). On the right: the cycle $w = acaacbaacbaba$ obtained by following a loop in $\mathbb Z^2$ in a configuration of $X_G$.
• Figure 5: Partial representation of two walks $p,q$ which differ by a square.

Theorems & Definitions (91)

• Theorem
• Conjecture 1.1
• Definition 2.1
• Definition 2.4
• Definition 2.9
• Remark 2.11
• Remark 2.12
• Remark 2.13
• Remark 2.16
• Definition 2.17