自由群移位系统的典型 sofic 熵与局部极限
Typical sofic entropy and local limits for free group shift systems
AI总结：
本文研究自由群移位系统不变测度的典型上下 sofic 熵，给出其等于退火熵的条件，并用于随机正则图 Gibbs 态局部极限及 Ising 模型熵相变。
摘要：
我们证明，对于自由群移位系统上的任意不变测度 $μ$，都存在两个数 $h^\flat \leq h^\sharp$，它们在某种意义上是典型的上、下 sofic 熵值。我们还给出一个条件，在该条件下 $h^\flat = h^\sharp = \mathrm{f}(μ)$，其中 $\mathrm{f}$ 是退火熵（也称为 f 不变量）。这可用于计算随机正则图序列上有限型 Gibbs 态的典型局部极限。作为例子，我们求出了 Ising 模型和 Potts 模型的典型局部极限。我们还表明，对于 Markov 链，Kesten–Stigum 第二特征值重构判据实际上意味着在典型 sofic 逼近上不存在好模型（即 $h^\sharp = -\infty$）。特别地，我们得到自由边界 Ising 态的典型熵 $h^\flat = h^\sharp$ 的精确值：在相互作用强度不超过重构阈值时，它等于退火熵 $\mathrm{f}$；此后它会突然降至 $-\infty$。
英文摘要：
We show that for any invariant measure $μ$ on a free group shift system, there are two numbers $h^\flat \leq h^\sharp$ which in some sense are the typical upper and lower sofic entropy values. We also give a condition under which $h^\flat = h^\sharp = \mathrm{f}(μ)$, where $\mathrm{f}$ is the annealed entropy (also called the f invariant). This can be used to compute typical local limits of finitary Gibbs states over sequences of random regular graphs. As examples, we work out typical local limits of the Ising and Potts models. We also show that, for Markov chains, the Kesten--Stigum second-eigenvalue reconstruction criterion actually implies there are no good models over a typical sofic approximation (i.e. $h^\sharp = -\infty$). In particular, we have an exact value for the typical entropy $h^\flat = h^\sharp$ of the free-boundary Ising state: it is equal to the annealed entropy $\mathrm{f}$ for interaction strengths up to the reconstruction threshold, after which it drops abruptly to $-\infty$.
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