Gibbs measures for the fertile three-state hard-core models on a Cayley tree
This paper studies translation-invariant splitting Gibbs measures (TISGMs) for the fertile three-state hard-core models with activity \(\lambda>0\) on the Cayley tree of order \(k \geq 1\). There are four such models: wrench, wand, hinge, and pipe, which arise as simple examples of loss networks with nearest-neighbor exclusion. Some rigorous results have been obtained before for these models concerning the uniqueness and non-uniqueness of TISGMs in different parameter regions. This paper generalizes the result for hinge and wand cases and finds the exact critical value \(\lambda_{\mathrm{cr}}(k)\) for any \(k\geq 2\) such that for \(\lambda<\lambda_{\mathrm{cr}}\) there exists a unique TISGM and for \(\lambda>\lambda_{\mathrm{cr}}\) there exist multiple TISGMs. The authors also investigate sufficient conditions for the activity parameter \(\lambda\) to see extremality and non-extremality of the TISGM. Explicit formulae and some numerical values are given for the Cayley tree of order two.
scientific article; zbMATH DE number 6487391