The Blume-Emery-Griffiths model at the FAD and AD interfaces

TL;DR

This paper rigorously analyzes the Blume-Emery-Griffiths model at specific interfaces, using coupling and sampling methods to determine magnetization behavior and correlations in different dimensions.

Contribution

It introduces a Gibbs sampler for ground states, proves the existence and uniqueness of the zero-temperature Gibbs measure, and improves conditions for magnetization vanishing at low temperatures.

Findings

Non-zero spontaneous magnetization in 3D at zero temperature.

Vanishing magnetization in 2D with boundary conditions.

Exponential decay of correlations in 2D at zero temperature.

Abstract

We analyse the Blume-Emery-Griffiths (BEG) model on the lattice $\Zd$ at the ferromagnetic-antiquadrupolar-disordered (FAD) and antiquadrupolar-disordered (AD) interfaces of parameters. In our analysis of the FAD interface we introduce a Gibbs sampler of the ground states at zero temperature, and we exploit it in two different ways: first, we perform via perfect sampling an empirical evaluation of the spontaneous magnetization at zero temperature, finding a non-zero value in $d=3$ and a vanishing value in $d=2$. Second, using a careful coupling with the Bernoulli site percolation model in $d=2$, we prove rigorously that imposing $+$ boundary conditions, the magnetization in the center of a square box tends to zero in the thermodynamical limit and the two-point correlations decay exponentially. Also, using again a coupling argument, we show that the infinite volume Gibbs measure of the zero-temperature BEG exists and it is unique. In our analysis of the AD interface we restrict ourselves to $d=2$ and, by comparing the BEG model with a Bernoulli site percolation in a matching graph of $\mathbb{Z}^2$, we get a condition for the vanishing of the infinite volume limit magnetization improving, for low temperatures, earlier results obtained via expansion techniques.