Quantified block gluing for multidimensional subshifts of finite type: aperiodicity and entropy

Abstract

It is possible to extend the notion of block gluing for subshifts studied in [PS15] adding a gap function which gives the distance which allows to concatenate two rectangular blocks of the language. In this article, we study the interplay between this intensity and computational properties. In particular, we prove that there exists block gluing SFTs with linear gap which are aperiodic and that all the non-negative right-recursively enumerable ( 1 -computable) numbers can be realized as entropy of such subshifts of finite type. As block gluing with linear gap implies transitivity, this last point provides a solution to Problem 9.1 in [HM10] about the characterization of the entropies of transitive subshift of finite type.

A complete proof that square ice entropy is

“…This text is meant as a basis for further research that would aim at extending the computation method that we exposed to a broader set of multidimensional SFT, including, for instance, Kari–Culik tilings [C96], the monomer–dimer model [see, for instance, [FP05]], subshifts of square ice [GS17], the hard square shift [P12] or a three-dimensional version of the six-vertex model. Adaptations for these models may be possible, but would not be immediate at all.…”

“…Outside of the class of SFT defined by these symmetry restrictions, as far as we know, only strong mixing or measure theoretic conditions ensure algorithmic computability of entropy, leading, for instance, to relatively efficient algorithms approximating the hard square shift entropy [P12]. However, the irreducibility of the matrices derives from the irreducibility property of the stripes subshifts [Definition 2], that can be derived from the linear block gluing property of [GS17]. This property consists in the possibility for any pair of patterns on to be glued in any relative positions, provided that the distance between the two patterns is greater than a minimal distance, which is .…”

“…Some current research attempts have been made to understand the frontier between the uncomputability and the computability of entropy for a multidimensional SFT. For instance, approaching the frontier from the uncomputable domain, the author, together with Sablik [GS17], proved that the characterization of Hochman and Meyerovitch stands under a relaxed form of the constraint studied in [PS15], which includes notably all exactly solvable models considered in statistical and quantum physics. To approach the frontier from the computable domain, it is natural to attempt to understand (in particular, prove) and extend the computation methods developed for these models.…”

Parametrization by Horizontal Constraints in the Study of Algorithmic Properties of $\mathbb{Z}^2$-Subshift of Finite Type

“…It is possible to extend the notion of block gluingness by adding a gap function f that yields the distance f (n) which allows for the concatenation of two rectangular blocks of size n of the language: depending on the asymptotic behavior of f (n), the set of entropies can be either any Π 1 -computable number or only some computable numbers [Gd19]. The exact frontier for this parametrization is only known for subshifts with decidable language [GS20].…”

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

“…Edge of uncomputability. Together with M.Sablik [GS21], the first author proposed a quantification of the block gluing property, in which a function of the size of the patterns, called gap function, replaces the constant. This function reflects the 'range of order' in the system: the larger this function, the farther the presence of one particular pattern has an influence over the configuration.…”