In a recent paper, Giardinà et al. [ALEA Lat. Am. J. Probab. Math. Stat. 13 (2016) 121–161] have proved a law of large number and a central limit theorem with respect to the annealed measure for the magnetization of the Ising model on some random graphs, including the random 2-regular graph. In this paper, we present a new proof of their results which applies to all random regular graphs. In addition, we prove the existence of annealed pressure in the case of configuration model random graphs.
"Annealed limit theorems for the Ising model on random regular graphs." Ann. Appl. Probab. 29 (3) 1398 - 1445, June 2019. https://doi.org/10.1214/17-AAP1377