A Three State Hard-Core Model on a Cayley Tree

DOI

10.2991/jnmp.2005.12.3.7How to use a DOI?

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

© 2006, the Authors. Published by Atlantis Press.

Open Access

This is an open access article distributed under the CC BY-NC license (http://creativecommons.org/licenses/by-nc/4.0/).

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TY  - JOUR
AU  - James Martin
AU  - Utkir Rozikov
AU  - Yuri Suhov
PY  - 2005
DA  - 2005/08/01
TI  - A Three State Hard-Core Model on a Cayley Tree
JO  - Journal of Nonlinear Mathematical Physics
SP  - 432
EP  - 448
VL  - 12
IS  - 3
SN  - 1776-0852
UR  - https://doi.org/10.2991/jnmp.2005.12.3.7
DO  - 10.2991/jnmp.2005.12.3.7
ID  - Martin2005
ER  -