Atlantis Press Paper Details: A Three State Hard-Core Model on a Cayley Tree

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 (x) at each node x of the Cayley tree can be 0, 1 and 2. We have Poisson flow of calls of rate 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, > 0 and k 1, there exists a unique translatioinvariant splitting Gibbs measure µ . 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.

		back to table of contents

title:		A Three State Hard-Core Model on a Cayley Tree
publication:		JNMP
volume-issue:		12 - 3
pages:		432 - 448
ISSN:		1402-9251
DOI:		doi:10.2991/jnmp.2005.12.3.7 (how to use a DOI)
author(s):		James MARTIN, Utkir ROZIKOV, Yuri SUHOV
publication date:		August 2005
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 (x) at each node x of the Cayley tree can be 0, 1 and 2. We have Poisson flow of calls of rate 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, > 0 and k 1, there exists a unique translatioinvariant splitting Gibbs measure µ . 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.
copyright:		© Atlantis Press. This article is distributed under the terms of the Creative Commons Attribution License, which permits non-commercial use, distribution and reproduction in any medium, provided the original work is properly cited.
full text:		123art4.pdf (190 K)