### 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 λ_{c} = 1/(k - 1) × (k/(k-1)^{k}. Then: (i) for λ ≥ λ_{c}, the Gibbs measure is unique (and coincides with the above measure μ*), (ii) for λ > λ_{c}, 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 > λ_{c}. 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, λ_{e} and λ_{o}, for even and odd sites. We discuss open problems and state several related conjectures.

Original language | English (US) |
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Pages (from-to) | 197-212 |

Number of pages | 16 |

Journal | Queueing Systems |

Volume | 46 |

Issue number | 1-2 |

DOIs | |

State | Published - 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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## Cite this

*Queueing Systems*,

*46*(1-2), 197-212. https://doi.org/10.1023/b:ques.0000021149.43343.05