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On independence and entropy for high‐dimensional isotropic subshifts
Author(s) -
Meyerovitch Tom,
Pavlov Ronnie
Publication year - 2014
Publication title -
proceedings of the london mathematical society
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.899
H-Index - 65
eISSN - 1460-244X
pISSN - 0024-6115
DOI - 10.1112/plms/pdu029
Subject(s) - mathematics , entropy (arrow of time) , isotropy , combinatorics , limiting , discrete mathematics , bernoulli's principle , entropy rate , pure mathematics , joint quantum entropy , principle of maximum entropy , statistics , quantum mechanics , physics , engineering , aerospace engineering , mechanical engineering
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 ) log k 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.