Approximating the hard square entropy constant with probabilistic methods

Abstract

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.