APPROXIMATING THE HARD SQUARE ENTROPY CONSTANT WITH PROBABILISTIC METHODS

• 成果类型: Article
• 署名作者: Pavlov, Ronnie
• 署名单位: University of British Columbia
• 刊物名称: ANNALS OF PROBABILITY
• ISSN/ISSBN: 0091-1798
• DOI: 10.1214/11-AOP681
• 发表日期: 2012
• 页码: 2362-2399
• 关键词: finite-type; subshifts; shifts; percolation; lattice; number; morphisms

摘要

For any two-dimensional nearest neighbor shift of finite type X and any integer n >= 1, one can define the horizontal strip shift H-n (X) to be the set of configurations on Z x {1, ... , n} which do not contain any forbidden pairs of adjacent letters for X. It is always the case that the sequence h(top)(H-n(X))/n of normalized topological entropies of the strip shifts converges to h(toP)(X), the topological entropy of X. In this paper, we combine ergodic theoretic techniques with methods from percolation theory and interacting particle systems to show that for the two-dimensional hard square shift H, the sequence h(top)(Hn+1(H)) - h(top)(H-n(H)) also converges to h(top)(H), and that the rate of convergence is at least exponential. As a corollary, we show that h(top)(H) is computable to any tolerance epsilon in time polynomial in 1/epsilon. We also show that this phenomenon is not true in general by defining a block gluing two-dimensional nearest neighbor shift of finite type Y for which h(top)(Hn+1(Y)) - h(top)(H-n(Y)) does not even approach a limit.