Dilution robustness for mean field ferromagnets

In this work we compare two different random dilutions on a mean field ferromagnet: the first model is built on a Bernoulli diluted graph while the second lives on a Poisson diluted one. While it is known that the two models have, in the thermodynamic limit, the same free energy, we investigate the structural constraints that the two models must fulfill. We rigorously derive for each model the set of identities for the multi-overlap distribution, using different methods for the two dilutions: constraints in the former model are obtained by studying the consequences of the self-averaging of the internal energy density, while in the latter they are obtained by a stochastic stability technique. Finally we prove that the identities emerging in the two models are the same, showing robustness of the ferromagnetic properties of diluted graphs with respect to the details of dilution.