Large deviations and law of large numbers for a mean field type interacting particle systems

In this paper, we are interested in the behaviour of the empirical measure of a large exchangeable system of N interacting particles living in a Polish space S, whose law in SN is the Gibbs measure given at 0.1. We get a Sanov type result for the large deviations of the empirical measure, and a weak law of large numbers, as N tends to infinity. Both handle the case of phase coexistence. A strong law of large numbers is obtained when the infinite system has a unique phase. All these convergences take place in a subspace of the probability measures whose topology is at least stronger than the usual weak one.