On independence and entropy for high-dimensional isotropic subshifts

TL;DR

This paper derives an explicit formula for the asymptotic growth rate of high-dimensional array configurations avoiding certain patterns, linking it to a one-dimensional entropy concept called independence entropy, with implications for measure limits and existing models.

Contribution

It introduces the independence entropy as a computable expression for the limiting entropy of high-dimensional isotropic subshifts, generalizing and unifying previous results.

Findings

Derived an explicit formula for limiting entropy involving only one-dimensional words.

Characterized weak limits of measures of maximal entropy as Bernoulli extensions.

Applied results to recover and extend known models in the literature.

Abstract

In this work, we study the problem of finding the asymptotic growth rate of the number of of $d$-dimensional arrays with side length $n$ over a given alphabet which avoid a list of one-dimensional "forbidden" words along all cardinal directions, as both $n$ and $d$ tend to infinity. Louidor, Marcus, and the second author called this quantity the "limiting entropy"; it is the limit of a sequence of topological entropies of a sequence of isotropic $\mathbb{Z}^d$ subshifts with the dimension $d$ tending to infinity. We find an expression for this limiting entropy which involves only one-dimensional words, which was implicitly conjectured earlier, and given the name "independence entropy." In the case where the list of "forbidden" words is finite, this expression is algorithmically computable and is of the form $\frac{1}{n} \log k$ for $k,n \in \mathbb{N}$. Our proof also characterizes the weak limits (as $d \rightarrow \infty$) of isotropic measures of maximal entropy; any such measure is a Bernoulli extension over some zero entropy factor from an explicitly defined set of measures. We also demonstrate how our results apply to various models previously studied in the literature, in some cases recovering or generalizing known results, but in other cases proving new ones.