The Blume–Emery–Griffiths Model on the FAD Point and on the AD Line

We analyse the Blume–Emery–Griffiths (BEG) model on the lattice \({\mathbb {Z}}^d\) on the ferromagnetic-antiquadrupolar-disordered (FAD) point and on the antiquadrupolar-disordered (AD) line. In our analysis on the FAD point, 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 under 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 there exists a unique zero-temperature infinite-volume Gibbs measure for the BEG. In our analysis of the AD line 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.