Existence of phase transitions near the displacive limit of a classicaln-component lattice model

Using a new method proposed by Fröhlich, Simon, and Spencer, we prove the existence of a phase transition near the displacive limit of a classical $n$-component displacement model on a $d$-dimensional ($d\ge 3$) lattice. In certain cases, the proof can be extended for $d=2\mathrm{}$ and $n=1\mathrm{}$. Moreover, we derive exact lower bounds for the critical temperature of the spin-$s$ ($s=\frac{1}{2},1,\frac{3}{2},\dots$) extension of the Blume-Capel model.