On local equilibrium equations for clustering states

In this note we show that local equilibrium equations (the generalization of the TAP equations or of the belief propagation equations) do have solutions in the colorable phase of the coloring problem. The same results extend to other optimization problems where the solutions has cost zero (e.g. K-satisfiability). On a random graph the solutions of the local equilibrium equations are associated to clusters of configurations (clustering states). On a random graph the local equilibrium equations have solutions almost everywhere in the uncolored phase; in this case we have to introduce the concept quasi-solution of the local equilibrium equations.