A hard-core model on a Cayley tree: An example of a loss network

Y. Suhov, U. A. Rozikov

Research output: Contribution to journalArticle

38 Scopus citations

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 λ > 1/(√k - 1) × (√k/√k - 1))k, measure μ* is not extreme (this result can be improved). Finally, we consider a model with two fugacities, λe and λo, for even and odd sites. We discuss open problems and state several related conjectures.

Original languageEnglish (US)
Pages (from-to)197-212
Number of pages16
JournalQueueing Systems
Volume46
Issue number1-2
DOIs
StatePublished - 2004

All Science Journal Classification (ASJC) codes

  • Statistics and Probability
  • Computer Science Applications
  • Management Science and Operations Research
  • Computational Theory and Mathematics

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