Quenched Central Limit Theorems for the Ising Model on Random Graphs

Themain goal of the paper is to prove central limit theorems for the magnetization
rescaled by the square root of 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 β > βc
or β > 0 and B not equal to 0, where β is the inverse temperature, βc 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 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