Short-range and long-range order: a transition in block-gluing behavior in Hom shifts - Laboratoire Interdisciplinaire des Sciences du Numérique Access content directly
Journal Articles Journal d'analyse mathématique Year : 2024

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 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.
Fichier principal
Vignette du fichier
Homshifts.pdf (1.01 Mo) Télécharger le fichier
Origin : Files produced by the author(s)

Dates and versions

hal-03842725 , version 1 (07-11-2022)
hal-03842725 , version 2 (24-01-2024)

Identifiers

  • HAL Id : hal-03842725 , version 2

Cite

Silvère Gangloff, Benjamin Hellouin de Menibus, Piotr Opocha. Short-range and long-range order: a transition in block-gluing behavior in Hom shifts. Journal d'analyse mathématique, inPress. ⟨hal-03842725v2⟩
64 View
17 Download

Share

Gmail Facebook X LinkedIn More