ISING MODELS ON LOCALLY TREE-LIKE GRAPHS
ISING MODELS ON LOCALLY TREE-LIKE GRAPHS
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DOI：
10.1214/09-aap627
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发表时间：
2010-04-01
影响因子：
1.8
通讯作者：
Montanari, Andrea
中科院分区：
文献类型：
--
作者：
Dembo, Amir;Montanari, Andrea
我们认为铁磁伊辛模型的图，局部收敛到树。例子包括有界度的随机正则图和有界平均度的一致随机图。我们证明了对于任何正温度和外场，“腔”预测的每自旋极限自由能是正确的。此外，局部边缘可以通过迭代一组平均场（腔）方程来近似。这两个结果都是通过证明原始图上的玻尔兹曼分布局部收敛到适当的无限随机树上的玻尔兹曼分布来实现的。
We consider ferromagnetic Ising models on graphs that converge locally to trees. Examples include random regular graphs with bounded degree and uniformly random graphs with bounded average degree. We prove that the "cavity" prediction for the limiting free energy per spin is correct for any positive temperature and external field. Further, local marginals can be approximated by iterating a set of mean field (cavity) equations. Both results are achieved by proving the local convergence of the Boltzmann distribution on the original graph to the Boltzmann distribution on the appropriate infinite random tree.