Multiplicity of Phase Transitions and Mean-Field Criticality on Highly Non-Amenable Graphs
Multiplicity of Phase Transitions and Mean-Field Criticality on Highly Non-Amenable Graphs
复制标题
高度不适宜图上的相变多重性和平均场临界性
DOI：
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发表时间：
2001
期刊：
影响因子：
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通讯作者：
R. Schonmann
中科院分区：
文献类型：
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作者：
R. Schonmann
摘要：我们考虑独立渗流，Ising和Potts模型，和接触过程，在无限的，局部有限的，连通graphs.它表明，在图的边等周Cheeger常数足够大，在度的图的顶点，每个模型表现出一个以上的临界点，分离定性不同的制度。对于这类幺模传递图，证明了独立渗流、Ising模型和接触过程的临界行为是平均场型的;对于幺模传递图上的Potts模型，证明了自由Gibbs测度在自同构不变Gibbs测度集中是极值的性质在温度下的单调性，并证明了相应的临界温度为正的充要条件是图上独立键渗流的无限团簇的唯一性阈值小于1.我们建立了有限团簇的临界温度为正的条件，在大密度下独立渗流岛性质，并使用这些来表明，对于一大类图，q状态Potts模型具有低温区域，其中自由吉布斯测度分解为q有序相的均匀混合物。有一端的顺从传递平面图，我们表明，q状态Potts模型有一个临界点，分离的制度，高温中的自由吉布斯措施是极值的自同构-不变的吉布斯措施从一个政权的低温，其中自由吉布斯措施分解为均匀混合物的q有序相。
Abstract: We consider independent percolation, Ising and Potts models, and the contact process, on infinite, locally finite, connected graphs.It is shown that on graphs with edge-isoperimetric Cheeger constant sufficiently large, in terms of the degrees of the vertices of the graph, each of the models exhibits more than one critical point, separating qualitatively distinct regimes. For unimodular transitive graphs of this type, the critical behaviour in independent percolation, the Ising model and the contact process are shown to be mean-field type.For Potts models on unimodular transitive graphs, we prove the monotonicity in the temperature of the property that the free Gibbs measure is extremal in the set of automorphism invariant Gibbs measures, and show that the corresponding critical temperature is positive if and only if the threshold for uniqueness of the infinite cluster in independent bond percolation on the graph is less than 1.We establish conditions which imply the finite-island property for independent percolation at large densities, and use those to show that for a large class of graphs the q-state Potts model has a low temperature regime in which the free Gibbs measure decomposes as the uniform mixture of the q ordered phases.In the case of non-amenable transitive planar graphs with one end, we show that the q-state Potts model has a critical point separating a regime of high temperatures in which the free Gibbs measure is extremal in the set of automorphism-invariant Gibbs measures from a regime of low temperatures in which the free Gibbs measure decomposes as the uniform mixture of the q ordered phases.