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

TL;DR

This paper investigates the transition in the behavior of the gluing gap in Hom shifts, revealing a dichotomy between linear and logarithmic growth, and provides a counterexample to a previous conjecture.

Contribution

It proves that the gluing gap in Hom shifts is either linear or logarithmic in size, and constructs a shift with a logarithmic gap, challenging prior assumptions.

Findings

Gluing gap either depends linearly on n or is dominated by log(n)

Constructed a Hom shift with a gap of order log(n)

Disproved a conjecture by Pavlov and Schraudner

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 {\Theta}(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.