Rates of Convergence in the Blume–Emery–Griffiths Model

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.