Splitting Gibbs Measures Research Articles

Gibbs measures for the SOS model with four states on a Cayley tree

We analyze the SOS (solid-on-solid) model with spins 0, 1, 2, 3 on a Cayley tree of order k ≥ 1. We consider translation-invariant and periodic splitting Gibbs measures for this model. The majority of the constructed Gibbs measures are mirror symmetric.

The Potts Model with Countable Set of Spin Values on a Cayley Tree

We consider a nearest-neighbor Potts model, with countable spin values 0,1,..., and non zero external field, on a Cayley tree of order k (with k+1 neighbors). We study translation-invariant ‘splitting’ Gibbs measures. We reduce the problem to the description of the solutions of some infinite system of equations. For any k≥ 1 and any fixed probability measure ν with ν (i)>0 on the set of all non negative integer numbers Φ={0,1,...} we show that the set of translation-invariant splitting Gibbs measures contains at most one point, independently on parameters of the Potts model with countable set of spin values on Cayley tree. Also we give description of the class of measures ν on Φ such that with respect to each element of this class our infinite system of equations has unique solution {ai, i=1,2,...}, where a ∈(0,1).

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 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.

A Hard-Core Model on a Cayley Tree: An Example of a Loss Network

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 kg1, there exists a unique translation-invariant splitting Gibbs measure μ*. Define λce1/(k−1)×(k/(k−1))k. Then: (i) for λlλc, 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/($\sqrt{k}$−1)×$\sqrt{k}$/$\sqrt{k}$−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.