### 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) |
---|---|

Pages (from-to) | 197-212 |

Number of pages | 16 |

Journal | Queueing Systems |

Volume | 46 |

Issue number | 1-2 |

State | Published - Jan 1 2004 |

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### All Science Journal Classification (ASJC) codes

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

### Cite this

*Queueing Systems*,

*46*(1-2), 197-212.

}

*Queueing Systems*, vol. 46, no. 1-2, pp. 197-212.

**A hard-core model on a Cayley tree : An example of a loss network.** / Soukhov, Iouri M.; Rozikov, U. A.

Research output: Contribution to journal › Article

TY - JOUR

T1 - A hard-core model on a Cayley tree

T2 - An example of a loss network

AU - Soukhov, Iouri M.

AU - Rozikov, U. A.

PY - 2004/1/1

Y1 - 2004/1/1

N2 - 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.

AB - 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.

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UR - http://www.scopus.com/inward/citedby.url?scp=3543148922&partnerID=8YFLogxK

M3 - Article

AN - SCOPUS:3543148922

VL - 46

SP - 197

EP - 212

JO - Queueing Systems

JF - Queueing Systems

SN - 0257-0130

IS - 1-2

ER -