TY - JOUR

T1 - A three state hard-core model on a cayley tree

AU - Martin, James

AU - Rozikov, Utkir

AU - Suhov, Yuri

N1 - Copyright:
Copyright 2017 Elsevier B.V., All rights reserved.

PY - 2005

Y1 - 2005

N2 - 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.

AB - 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.

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U2 - 10.2991/jnmp.2005.12.3.7

DO - 10.2991/jnmp.2005.12.3.7

M3 - Article

AN - SCOPUS:25444445015

VL - 12

SP - 432

EP - 448

JO - Journal of Nonlinear Mathematical Physics

JF - Journal of Nonlinear Mathematical Physics

SN - 1402-9251

IS - 3

ER -