Solution of the Monomer-Dimer Model on Locally Tree-Like Graphs. Rigorous Results

We consider the monomer-dimer model on sequences of random graphs locally convergent to trees. We prove that the monomer density converges almost surely, in the thermodynamic limit, to an analytic function of the monomer activity. We characterise this limit as the expectation of the solution of a fixed point distributional equation and we give an explicit expression of the limiting pressure per particle.