We perform a detailed study of the critical behavior of the mean field diluted Ising ferromagnet by analytical and numerical tools. We obtain self-averaging for the magnetization and write down an expansion for the free energy close to the critical line. The scaling of the magnetization is also rigorously obtained and compared with extensive Monte Carlo simulations. We explain the transition from an ergodic region to a non trivial phase by commutativity breaking of the infinite volume limit and a suitable vanishing field. We find full agreement among theory, simulations and previous results.

A relation between the effective diffusion coefficient in a lattice with random site energies and random transition rates and the macroscopic conductivity in a random resistor network allows for elucidating possible sources of anomalous diffusion in random potential models. We show that subdiffusion is only possible either if the mean Boltzmann factor in the corresponding potential diverges or if the percolation concentration in the system is equal to unity (or both), and that superdiffusion is impossible in our system under any condition. We show also other useful applications of this relation.

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 on the structural constraints that the two models must fulfill. We rigorously derive for each model the set of identities for the multi-overlaps 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 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.

In this paper we develop the interpolating cavity eld technique for the mean eld ferromagnetic p-spin. The model we introduce is a natural extension of the diluted Curie-Weiss model to p > 2 spin interactions. Several properties of the free energy are analyzed and, in particular, we show that it recovers the expressions already known for p = 2 models and for p > 2 fully connected models. Further, as the model lacks criticality, we present extensive numerical simulations to evidence the presence of a rst order phase transition and deepen the behavior at the transition line. Overall, a good agreement is obtained among analytical results, numerics and previous works.

We analyze diffusion of small particles in a solid polymeric medium taking into account a short-range particle-polymer interaction. The system is modeled by a particle diffusion on a ternary lattice where the sites occupied by polymer segments are blocked, the ones forming the hull of the chains correspond to the places at which the interaction takes place, and the rest are voids, in which the diffusion is free. In the absence of interaction the diffusion coefficient shows only a weak dependence on the polymer chain length and its behavior strongly resembles the usual site percolation. In the presence of interactions the diffusion coefficient (and especially its temperature dependence) shows a nontrivial behavior depending on the sign of interaction and on whether the voids and the hulls of the chains percolate or not. The temperature dependence may be Arrhenius-like or strongly non-Arrhenius, depending on parameters. The analytical results obtained within the effective medium approximation are in qualitative agreement with those of Monte Carlo simulations.