The Annals of Probability

Approximating the hard square entropy constant with probabilistic methods

Ronnie Pavlov

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For any two-dimensional nearest neighbor shift of finite type $X$ and any integer $n\geq1$, one can define the horizontal strip shift $H_{n}(X)$ to be the set of configurations on $\mathbb{Z}\times\{1,\ldots,n\}$ which do not contain any forbidden pairs of adjacent letters for $X$. It is always the case that the sequence $h^{\mathrm{top} }(H_{n}(X))/n$ of normalized topological entropies of the strip shifts converges to $h^{\mathrm{top} }(X)$, 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 $\mathcal{H}$, the sequence $h^{\mathrm{top} }(H_{n+1}(\mathcal{H}))-h^{\mathrm{top} }(H_{n}(\mathcal{H}))$ also converges to $h^{\mathrm{top} }(\mathcal{H})$, and that the rate of convergence is at least exponential. As a corollary, we show that $h^{\mathrm{top} }(\mathcal{H})$ is computable to any tolerance $\varepsilon$ in time polynomial in $1/\varepsilon$. 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 $h^{\mathrm{top} }(H_{n+1}(Y))-h^{\mathrm{top} }(H_{n}(Y))$ does not even approach a limit.

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Ann. Probab., Volume 40, Number 6 (2012), 2362-2399.

First available in Project Euclid: 26 October 2012

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Zentralblatt MATH identifier

Primary: 37B50: Multi-dimensional shifts of finite type, tiling dynamics
Secondary: 37B40: Topological entropy 82B20: Lattice systems (Ising, dimer, Potts, etc.) and systems on graphs 60K35: Interacting random processes; statistical mechanics type models; percolation theory [See also 82B43, 82C43]

$\mathbb{Z}^{d}$ SFT entropy stochastic dominance percolation


Pavlov, Ronnie. Approximating the hard square entropy constant with probabilistic methods. Ann. Probab. 40 (2012), no. 6, 2362--2399. doi:10.1214/11-AOP681.

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