A Hard-Core Model on a Cayley Tree: An Example of a Loss Network |
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Authors: | Y Suhov UA Rozikov |
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Institution: | 1. Statistical Laboratory, DPMMS, University of Cambridge, Cambridge, CB3 0WB, UK 2. Institute of Mathematics, Uzbek Academy of Sciences, Tashkent, 700143, Uzbekistan
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Abstract: | The paper is about a nearest-neighbor hard-core model, with fugacity λ>0, on a homogeneous Cayley tree of order k(with k+1 neighbors). This model arises as as a simple example of a loss network with a nearest-neighbor exclusion. We focus on Gibbs measures for the hard core model, in particular on ‘splitting’ Gibbs measures generating a Markov chain along each path on the tree. In this model, ?λ>0 and k≥1, there exists a unique translation-invariant splitting Gibbs measure μ*. Define λc=1/(k?1)×(k/(k?1)) k . Then: (i) for λ≤λc, the Gibbs measure is unique (and coincides with the above measure μ*), (ii) for λ>λc, in addition to μ*, there exist two distinct translation-periodic measures, μ+and μ?, taken to each other by the unit space shift. Measures μ+and μ?are extreme ?λ>λc. We also construct a continuum of distinct, extreme, non-translational-invariant, splitting Gibbs measures. For $\lambda >1/(\sqrt k - 1) \times (\sqrt k /\sqrt k - 1))^k $ , measure μ*is not extreme (this result can be improved). Finally, we consider a model with two fugacities, λeand λo, for even and odd sites. We discuss open problems and state several related conjectures. |
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