Phase transitions for 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 $β$ 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. We observe an additional phase transition phenomenon in the sense that depending on how fast $K_n$ and $β_n$ are converging to points in various subsets of the phase diagram, different rates of convergences to one and the same limiting distribution occur.