Four-Cycle Free Graphs, Height Functions, the Pivot Property and Entropy Minimality

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.

Square-free graphs are multiplicative

“…An algorithm for K = K p/q for 2 ≤ p q < 4 was recently shown by Brewster et al [Bre+15]. Interesting, partly related properties of homomorphisms from infinite grids Z d to square-free graphs, with motivations in statistical thermodynamics, were found independently by Chandgotia [Cha14].…”