Statistical Mechanics of Hard-Core Particles with Attractive Interactions

Mean-field monomer-dimer models, on sparse random graphs or on the complete graph, can be considered as an approximation of finite-dimensional physical models involving particles of different sizes. On the other hand they have a particular interest for the emerging applications to Computer Science and Social Sciences, since the real-world networks are often modelled by particular families of random graphs. We give a rigorous proof of Zdeborovà-Mézard's picture of the monomer-dimer model with pure hard-core interaction on sparse random graphs. As shown by Heilmann and Lieb, the hard-core interaction is not sufficient to cause a phase transition in monomer-dimer models. We study monomer-dimer models on the complete graph and in particular we add an attractive interaction to the hard-core one. We provide the solution of this model, showing that a phase transition occurs. The critical exponents are the standard mean-field ones and the central limit theorem breakdowns. Finite-dimensional monomer-dimer models (and more general hard-rods models) are still interesting also for applications to Physics, in the theory of liquid crystals. Heilmann and Lieb proposed some monomer-dimer models on Z^2 with attractive interactions that favour the presence of clusters of neighbouring parallel dimers. They showed the presence of orientational order at low temperatures, while they conjectured the absence of translational order. We prove the absence of translational order in a different framework, when the dimer potential favours one of the two orientations.