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Journal of Statistical Physics

, Volume 154, Issue 6, pp 1483–1507 | Cite as

Rates of Convergence in the Blume–Emery–Griffiths Model

  • Peter EichelsbacherEmail author
  • Bastian Martschink
Article

Abstract

We derive rates of convergence for limit theorems that reveal the intricate structure of the phase transitions in a mean-field version of the Blume–Emery–Griffith model. The theorems consist of scaling limits for the total spin. The model depends on the inverse temperature \(\beta \) and the interaction strength \(K\). The rates of convergence results are obtained as \((\beta ,K)\) converges along appropriate sequences \((\beta _n,K_n)\) to points belonging to various subsets of the phase diagram which include a curve of second-order points and a tricritical point. We apply Stein’s method for normal and non-normal approximation avoiding the use of transforms and supplying bounds, such as those of Berry–Esseen quality, on approximation error.

Keywords

Stein’s method Exchangeable pairs Blume–Emery–Griffith model  Second-order phase transition First-order phase transition Tricritical point  Blume–Capel model 

Mathematics Subject Classification

Primary 60F05 Secondary 82B20 82B26 

Notes

Acknowledgments

The authors have been supported by Deutsche Forschungsgemeinschaft via SFB/TR 12.

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Copyright information

© Springer Science+Business Media New York 2014

Authors and Affiliations

  1. 1.Fakultät für MathematikRuhr-Universität BochumBochumGermany
  2. 2.Hochschule Bonn-Rhein SiegSankt AugustinGermany

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