  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.
