On independence and entropy for high-dimensional isotropic subshifts

In this work, we study the problem of finding the asymptotic growth rate of the number 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 ℤ^d subshifts with the dimension d tending to infinity ['Independence entropy of ℤ^d-shift spaces', Acta. Appl. Math. 126 (2013) 297-317]. We find an expression for this limiting entropy which involves only one-dimensional words, which was implicitly conjectured in [E. Louidor, B. Marcus and R. Pavlov, 'Independence entropy of ℤ^d-shift spaces', Acta. Appl. Math. 126 (2013) 297-317], 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 (1/n)logk for k, n ∈ ℕ. Our proof also characterizes the weak limits (as d → ∞) of isotropic measures of maximal entropy; any such measure is a Bernoulli extension over some zero entropy factor taken 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. The core idea of our proof is to consider certain isotropic measures on a limiting 'infinite-dimensional' subshift, and apply a variant of the classical theorem of de Finetti on exchangeable random variables.