Approximating the Hard Square Entropy Constant with Probabilistic Methods

TL;DR

This paper demonstrates that for the hard square shift, the difference in normalized topological entropies of successive horizontal strips converges exponentially to the overall entropy, enabling efficient approximation.

Contribution

It introduces probabilistic methods to show exponential convergence of entropy differences for the hard square shift, and provides a counterexample for a different shift.

Findings

Exponential convergence of h(H_{n+1}(H)) - h(H_n(H)) to h(H)

h(H) is computable in polynomial time with arbitrary precision

Counterexample shift Y where entropy differences do not converge

Abstract

For any two-dimensional nearest neighbor shift of finite type X and any integer n > 0, 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 transitions for X. It is always the case that the sequence h(H_n(X))/n of normalized topological entropies of the strip shifts approaches h(X), the topological entropy of X. In this paper, we use probabilistic methods from interacting particle systems to show that for the two-dimensional hard square shift H, in fact h(H_{n+1}(H)) - h(H_n(H)) also approaches h(H), and the rate of convergence is at least exponential. A consequence of this is that h(H) is computable to any tolerance 1/n in time polynomial in n. We also give an example of a two-dimensional block gluing nearest neighbor shift of finite type Y for which h(H_{n+1}(Y)) - h(H_n(Y)) does not even approach a limit.