Abstract
AbstractWe analyse the Blume–Emery–Griffiths (BEG) model on the lattice $${\mathbb {Z}}^d$$
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 $$d=3$$
d
=
3
and a vanishing value in $$d=2$$
d
=
2
. Second, using a careful coupling with the Bernoulli site percolation model in $$d=2$$
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 $$d=2$$
d
=
2
and, by comparing the BEG model with a Bernoulli site percolation in a matching graph of $${\mathbb {Z}}^2$$
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.
Funder
MIUR
Università degli Studi di Roma Tor Vergata
Publisher
Springer Science and Business Media LLC
Subject
Mathematical Physics,Statistical and Nonlinear Physics