We study translation-invariant Gibbs measures for the Blume–Capel model with wand on a Cayley tree of order k. We find the exact critical value 𝜃_cr = 1 such that, for 𝜃 ≥ 𝜃_cr , there exists a unique translation-invariant Gibbs measure and, for 0 < 𝜃 < 𝜃_cr , there are exactly three translation-invariant Gibbs measures in the presence of a wand in the analyzed model. In addition, we study the problem of (no) extremes for these measures.

中文翻译：

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在Cayley树上放弃的Blume-Capel模型的平移不变极端Gibbs测度

我们研究k阶Cayley树上带有魔杖的Blume-Capel模型的平移Gibbs测度。我们发现精确的临界值𝜃 _cr = 1，使得对于𝜃≥𝜃 _cr，存在一个唯一的平移不变吉布斯测度；对于0 <𝜃 <𝜃 _cr，存在三个正变不变式Gibbs测度分析模型中的魔杖 另外，我们研究了这些措施的极端问题。