Self-organization and associative memory: 3rd edition
Self-organization and associative memory: 3rd edition
A second threshold for the hard-core model on a Bethe lattice
Random Structures & Algorithms - Isaac Newton Institute Programme “Computation, Combinatorics and Probability”: Part I
Insensitivity and stability of random-access networks
Performance Evaluation
Equalizing throughputs in random-access networks
ACM SIGMETRICS Performance Evaluation Review
Achieving target throughputs in random-access networks
Performance Evaluation
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The paper is about a nearest-neighbor hard-core model, with fugacity λ0, 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, ∀λ0 and k⩾1, there exists a unique translation-invariant splitting Gibbs measure μ*. Define λc=1/(k−1)×(k/(k−1))k. Then: (i) for λ⩽λc, the Gibbs measure is unique (and coincides with the above measure μ*), (ii) for λλc, 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 ∀λλc. We also construct a continuum of distinct, extreme, non-translational-invariant, splitting Gibbs measures. For λ1/($\sqrt{k}$−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.