Rates of Convergence in the Blume–Emery–Griffiths Model

• Published in: Journal of statistical physics Vol. 154; no. 6; pp. 1483 - 1507
• Main Authors: Eichelsbacher, Peter, Martschink, Bastian
• Format: Journal Article
• Language: English
• Published: Boston Springer US 01.03.2014; Springer
• Subjects: Mathematical and Computational Physics; Physical Chemistry; Physics; Physics and Astronomy; Quantum Physics; Statistical Physics and Dynamical Systems; Theoretical; Second-order phase transition; 82B26; Stein’s method; Blume–Emery–Griffith model; First-order phase transition; Primary 60F05; Exchangeable pairs; Secondary 82B20; Tricritical point; Blume–Capel model
• ISSN: 0022-4715; 1572-9613
• DOI: 10.1007/s10955-014-0925-y

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 β and the interaction strength K . The rates of convergence results are obtained as ( β , K ) converges along appropriate sequences ( β 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.