  A periodic Ising model is one endowed with interactions that are invariant
under translations of members of a full-rank sublattice $\mathfrak{L}$ of
$\mathbb{Z}^2$. We give an exact, quantitative description of the critical
temperature, defined by the supreme of the temperatures at which the
spontaneous magnetization of a periodic, Ising ferromagnets is nonzero, as the
solution of a certain algebraic equation, namely, the condition that the
spectral curve of the corresponding dimer model on the Fisher graph has a real
node on the unit torus. A simple proof for the exponential decay of spin-spin
correlations above the critical temperature for the symmetric, periodic Ising
ferromagnet, as well as the exponential decay of the edge-edge correlations for
all non-critical edge weights of the dimer model on periodic Fisher graphs, is
obtained by our technique.
