## Abstract

We consider a nearest-neighbor hard-core model, with three states, on a homogeneous Cayley tree of order k (with k + 1 neighbors). This model arises as a simple example of a loss network with nearest-neighbor exclusion. The state σ(x) at each node x of the Cayley tree can be 0, 1 and 2. We have Poisson flow of calls of rate λ at each site x, each call has an exponential duration of mean 1. If a call finds the node in state 1 or 2 it is lost. If it finds the node in state 0 then things depend on the state of the neighboring sites. If all neighbors are in state 0, the call is accepted and the state of the node becomes 1 or 2 with equal probability 1/2. If at least one neighbor is in state 1, and there is no neighbor in state 2 then the state of the node becomes 1. If at least one neighbor is in state 2 the call is lost. We focus on ‘splitting’ Gibbs measures for this model, which are reversible equilibrium distributions for the above process. We prove that in this model, ∀ λ > 0 and k ≥ 1, there exists a unique translationinvariant splitting Gibbs measure ^{*}. We also study periodic splitting Gibbs measures and show that the above model admits only translation - invariant and periodic with period two (chess-board) Gibbs measures. We discuss some open problems and state several related conjectures.

Original language | English (US) |
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Pages (from-to) | 432-448 |

Number of pages | 17 |

Journal | Journal of Nonlinear Mathematical Physics |

Volume | 12 |

Issue number | 3 |

DOIs | |

State | Published - 2005 |

## All Science Journal Classification (ASJC) codes

- Statistical and Nonlinear Physics
- Mathematical Physics