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.
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Acknowledgments
We are grateful to Aernout van Enter, with whom we discussed some of the topics in this work, for useful suggestions. We thank Élie de Panafieu and Nicolas Broutin for sharing their preprint [2] prior to publication. We acknowledge financial support from the Italian Research Funding Agency (MIUR) through FIRB project “Stochastic processes in interacting particle systems: duality, metastability and their applications”, Grant No. RBFR10N90W. The work of RvdH is supported in part by the Netherlands Organisation for Scientific Research (NWO) through VICI Grant 639.033.806 and the Gravitation Networks grant 024.002.003.
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Giardinà, C., Giberti, C., van der Hofstad, R. et al. Quenched Central Limit Theorems for the Ising Model on Random Graphs. J Stat Phys 160, 1623–1657 (2015). https://doi.org/10.1007/s10955-015-1302-1
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DOI: https://doi.org/10.1007/s10955-015-1302-1