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• W2953209329 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 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." @default.
• W2953209329 created "2019-06-27" @default.
• W2953209329 creator A5023789848 @default.
• W2953209329 creator A5055883859 @default.
• W2953209329 date "2011-12-06" @default.
• W2953209329 modified "2023-10-14" @default.
• W2953209329 title "On independence and entropy for high-dimensional isotropic subshifts" @default.
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• W2953209329 hasPublicationYear "2011" @default.
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