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This article is concerned with the asymptotic behaviour of random vectors ina diluted ferromagnetic model. We consider a model introduced by Bovier &Gayrard (1993) with ferromagnetic interactions on a directed Erd\H{o}s-R\'enyirandom graph. Here, directed connections between graph nodes are uniformlydrawn at random with a probability p that depends on the number of nodes N andis allowed to go to zero in the limit. If $Np\longrightarrow\infty$ in thismodel, Bovier & Gayrard (1993) proved a law of large numbers almost surely, andKabluchko et al. (2020) proved central limit theorems in probability. Here, wegeneralise these results for $\beta<1$ in the regime $Np\longrightarrow\infty$.We show that all those random vectors on the spin configuration that have alimiting distribution under the Curie-Weiss model converge weakly towards thesame distribution under the diluted model, in probability on graphrealisations. This generalises various results from the Curie-Weiss model tothe diluted model. As a special case, we derive a law of large numbers andcentral limit theorem for two disjoint groups of spins.

Random vectors on the spin configuration of a Curie-Weiss model on Erdos-Renyi random graphs