Multiplicity of Phase Transitions and Mean-Field Criticality on Highly Non-Amenable Graphs |
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Authors: | Roberto H Schonmann |
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Institution: | (1) Department of Mathematics, UCLA, Los Angeles, CA 90095, USA. E-mail: rhs@math.ucla.edu, US |
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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.
Received: 27 March 2000 / Accepted: 7 December 2000 |
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