Amenability and Phase Transition in the Ising Model
Amenability and Phase Transition in the Ising Model
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伊辛模型中的适应性和相变
DOI：
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发表时间：
1997
期刊：
影响因子：
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通讯作者：
J. Steif
中科院分区：
文献类型：
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作者：
J. Jonasson;J. Steif
考虑具有一致有界度的无限连通图G上具有外场h和耦合常数J的Ising模型。我们证明了如果G是不可服从的，则伊辛模型对于某些h≠0和某些J<；∞表现出相变。另一方面，如果G是顺从的和准传递的，则h≠0不会发生相变。特别地，当且仅当群的一个(全部)Cayley图上的Ising模型对某些h≠0和某些J<；∞表现出相变时，群是不可服从的。
We consider the Ising model with external field h and coupling constant J on an infinite connected graph G with uniformly bounded degree. We prove that if G is nonamenable, then the Ising model exhibits phase transition for some h≠0 and some J<∞. On the other hand, if G is amenable and quasi-transitive, then phase transition cannot occur for h≠0. In particular, a group is nonamenable if and only if the Ising model on one (all) of its Cayley graphs exhibits a phase transition for some h≠0 and some J<∞.