Quenched Central Limit Theorems for the Ising Model on Random Graphs

Abstract

The main goal of the paper is to prove central limit theorems for the magnetization rescaled by $$\sqrt{N}$$ for the Ising model on random graphs with N vertices. Both random quenched and averaged quenched measures are considered. We work in the uniqueness regime $$\beta >\beta _c$$ or $$\beta >0$$ and $$B\ne 0$$ , where $$\beta $$ is the inverse temperature, $$\beta _c$$ is the critical inverse temperature and B is the external magnetic field. In the random quenched setting our results apply to general tree-like random graphs (as introduced by Dembo, Montanari and further studied by Dommers and the first and third author) and our proof follows that of Ellis in $$\mathbb {Z}^d$$ . For the averaged quenched setting, we specialize to two particular random graph models, namely the 2-regular configuration model and the configuration model with degrees 1 and 2. In these cases our proofs are based on explicit computations relying on the solution of the one dimensional Ising models.