论文标题
吉布斯（Gibbs
Gibbs measures for HC-model with a countable set of spin values on a Cayley tree
论文作者
论文摘要
在本文中，我们在订单$ k \ geq 2 $的Cayley树上使用可数的$ \ Mathbb z $研究HC模型。该模型由一组可数参数集（即活动函数$λ_i> 0 $，$ i \ in \ Mathbb z $）定义。获得了一个功能方程，该方程为有限维吉布斯分布提供了一致性条件。分析该方程式，获得以下结果： - 令$λ= \sum_iλ_i$。 对于$λ=+\ infty $，没有翻译不变的吉布斯度量（TIGM），没有两周期的吉布斯度量（TPGM）； - 对于$λ<+\ infty $，可以证明Tigm的独特性； - 令$λ_ {\ rm cr}（k）= \ frac {k^k} {（k-1）^{k+1}} $。如果$ 0 <λ\leqλ_{\ rm cr} $，则恰好有一个tpgm是tigm; - 对于$λ>λ_ {\ rm cr} $，恰好有三个tpgms，其中之一是提格。
In this paper, we study the HC-model with a countable set $\mathbb Z$ of spin values on a Cayley tree of order $k\geq 2$. This model is defined by a countable set of parameters (that is, the activity function $λ_i>0$, $i\in \mathbb Z$). A functional equation is obtained that provides the consistency condition for finite-dimensional Gibbs distributions. Analyzing this equation, the following results are obtained: - Let $Λ=\sum_iλ_i$. For $Λ=+\infty$ there are no translation-invariant Gibbs measures (TIGM) and no two-periodic Gibbs measures (TPGM); - For $Λ<+\infty$, the uniqueness of TIGM is proved; - Let $Λ_{\rm cr}(k)=\frac{k^k}{(k-1)^{k+1}}$. If $0<Λ\leqΛ_{\rm cr}$, then there is exactly one TPGM that is TIGM; - For $Λ>Λ_{\rm cr}$, there are exactly three TPGMs, one of which is TIGM.
