Phase transitions for rates of convergence in the Blume-Emery-Griffiths model

TL;DR

This paper analyzes the rates of convergence in the Blume-Emery-Griffiths model, revealing how phase transitions affect the speed of convergence in limit theorems for the total spin, using Stein's method.

Contribution

It introduces a detailed analysis of convergence rates near phase transition points in the mean-field BEG model, employing Stein's method without transforms.

Findings

Different convergence rates occur depending on how parameters approach phase transition points.

The study provides bounds on approximation errors, including Berry-Esseen type bounds.

Phase transition phenomena influence the speed of convergence to limiting distributions.

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