The hard-core model on random graphs revisited

We revisit the classical hard-core model, also known as independent set and dual to vertex cover problem, where one puts particles with a first-neighbor hard-core repulsion on the vertices of a random graph. Although the case of random graphs with small and very large average degrees respectively are quite well understood, they yield qualitatively different results and our aim here is to reconciliate these two cases. We revisit results that can be obtained using the (heuristic) cavity method and show that it provides a closed-form conjecture for the exact density of the densest packing on random regular graphs with degree K ≥ 20, and that for K > 16 the nature of the phase transition is the same as for large K. This also shows that the hard-code model is the simplest mean-field lattice model for structural glasses and jamming. Given a graph, how to put a large number of particles on the vertices avoiding any firstneighbor contact? This is the task one has to solve in the hard-core model (HC), also known as independent set (IS), or vertex cover (VC) when the meaning of covered/empty is switched. It is a NP-hard combinatorial optimization problem that has many applications such as scheduling problems [1], inference of phylogenetic trees [2] or in the communications [3] where one seeks for the minimal set of placed sensor devices in a service area so that the entire area is accessible. The hard-core model is defined as follows: We consider a graph G = (V,E) of size N = |V | and associate an occupation number σi ∈ {0, 1} to every vertex i ∈ V , where 0 stands for free and 1 for occupied. The Gibbs measure that corresponds to the hard-core model reads P ({σi}i=1,...,N ) = 1