Journal of Statistical Physics

, Volume 169, Issue 3, pp 480–503

# Critical Behavior of the Annealed Ising Model on Random Regular Graphs

• Van Hao Can
Article

## Abstract

In Giardinà et al. (ALEA Lat Am J Probab Math Stat 13(1):121–161, 2016), the authors have defined an annealed Ising model on random graphs and proved limit theorems for the magnetization of this model on some random graphs including random 2-regular graphs. Then in Can (Annealed limit theorems for the Ising model on random regular graphs, arXiv:1701.08639, 2017), we generalized their results to the class of all random regular graphs. In this paper, we study the critical behavior of this model. In particular, we determine the critical exponents and prove a non standard limit theorem stating that the magnetization scaled by $$n^{3/4}$$ converges to a specific random variable, with n the number of vertices of random regular graphs.

## Keywords

Ising model Random graphs Critical behavior Annealed measure

## Mathematics Subject Classification

05C80 60F5 82B20

## Notes

### Acknowledgements

We would like to thank the anonymous referees for their carefully reading and their valuable comments. This work is supported by the Vietnam National Foundation for Science and Technology Development (NAFOSTED) under Grant number 101.03–2017.01.

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