Fine Asymptotics of the Magnetization of the Annealed Dilute Curie-Weiss Model

Abstract

Abstract We consider the dilute Curie-Weiss model of size N , which is a generalization of the classical Curie-Weiss model in which the dependency structure between the spins is encoded not by the complete graph but by the (directed) Erdős-Rényi graph on N vertices in which every edge appears independently with probability p ( N ). In the high-temperature regime with an external magnetic field ( $$0<\beta <1,h\in \mathbb {R}$$ 0 < β < 1 , h ∈ R ) we prove for $$p^{3}N^{2}\rightarrow \infty $$ p 3 N 2 → ∞ sharp cumulant bounds for the magnetization under the annealed Gibbs measure, which imply a central limit theorem with rate, moderate and large deviation principles, a concentration inequality, a normal approximation with Cramér correction and mod-Gaussian convergence.