Remarks on the Limiting GIbbs States on a (d+1)-Tree
Remarks on the Limiting GIbbs States on a (d+1)-Tree
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发表时间：
1977
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通讯作者：
Y. Higuchi
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作者：
Y. Higuchi
本文研究了（d+l）-树上具有最近邻铁磁势的极限Gibbs态。普雷斯顿[1]给出了这些模型中给定相互作用的Gibbs态不唯一的充分必要条件（即相变发生的充分必要条件）。文[1]和[2]研究了可数树上的Gibbs态。我们的目标是通过改变边界条件得到几个极限吉布斯态。Spitzer [2]证明了（i）图同构下的每个极值吉布斯状态不变量都是他定义意义上的“马尔可夫链”（见定义1），（ii）对于任何给定的最近邻铁磁势，吉布斯状态之间最多有三个“马尔可夫链”。在第4节中，我们将证明，对于给定的相互作用，每一个吉布斯的“马尔可夫链”都是作为具有某些边界条件的相同相互作用的极限吉布斯状态而得到的。在第5节中，我们将给出限制吉布斯态的例子，使得在相应的边界条件下出现的上自旋的数量远远小于每个边界上的下自旋的数量，而原点处的自旋向上的概率大于1/2。在第6节中，我们将使用上面的例子给出几个极值吉布斯状态。
In this paper we investigate limiting Gibbs states with nearest neighbour ferromagnetic potentials on a (d+l)-tree. Preston [1] has got a necessary and sufficient condition for the non-uniqueness of the Gibbs states for a given interaction (i.e. the necessary and sufficient condition for the phase transition to occur) in these models. The Gibbs states on a countable tree are studied in [1] and [2]. Our aim is to obtain several limiting Gibbs states by changing boundary conditions. Spitzer [2] has shown that (i) every extremal Gibbs state invariant under graph isomorphisms is a "Markov chain" in the sense of his definition (see Definition 1), and (ii) there are at most three "Markov chains" among the Gibbs states for any given nearest neighbour ferromagnetic potential. In section 4 we will prove that every "Markov chain" which is Gibbsian for the given interaction is obtained as a limiting Gibbs state for the same interaction with certain boundary conditions. In section 5 we will give examples of limiting Gibbs states such that the number of up-spins appearing in the corresponding boundary conditions is much smaller than that of down-spins on every boundary, while the probability for the spin at the origin to be up is larger than 1/2. In section 6 we will give several extremal Gibbs states using above examples.