用概率方法逼近硬平方熵常数
Approximating the Hard Square Entropy Constant with Probabilistic Methods
摘要：
对于任何有限型X的二维最近邻移位和任何整数n > 0，可以将水平带移位H_n（X）定义为Z x {1，. n}，不包含X的任何禁止跃迁。带移位的归一化拓扑熵序列h（H_n（X））/n总是逼近X的拓扑熵h（X）。本文用相互作用粒子系统的概率方法证明了对于二维硬平方位移H，h（H_{n+1}（H））- h（H_n（H））也逼近h（H），并且其收敛速率至少是指数级的.这样的结果是h（H）可计算到n中时间多项式的任何容差1/n。本文还给出了一个二维有限Y型块胶合最近邻移位的例子，它的h（H_{n+1}（Y））- h（H_n（Y））甚至不接近极限.
英文摘要：
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.
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