  Reexamining algebraic curves found in the eight-vertex model, we propose an
asymptotic form of the correlation functions for off-critical systems
possessing rotational and mirror symmetries of the square lattice, i.e., the
$C_{4v}$ symmetry. In comparison with the use of the Ornstein-Zernike form, it
is efficient to investigate the correlation length with its directional
dependence (or anisotropy). We investigate the $Q$-state Potts model on the
square lattice. Monte Carlo (MC) simulations are performed using the
infinite-size algorithm by Evertz and von der Linden. Fitting the MC data with
the asymptotic form above the critical temperature, we reproduce the exact
solution of the anisotropic correlation length (ACL) of the Ising model ($Q=2$)
within a five-digit accuracy. For $Q=3$ and 4, we obtain numerical evidence
that the asymptotic form is applicable to their correlation functions and the
ACLs. Furthermore, we successfully apply it to the bond percolation problem
which corresponds to the $Q\rightarrow1$ limit. From the calculated ACLs, the
equilibrium crystal shapes (ECSs) are derived via duality and Wulff's
construction. Regarding $Q$ as a continuous variable, we find that the ECS of
the $Q$-state Potts model is essentially the same as those of the Ising models
on the Union Jack and 4-8 lattices, which are represented in terms of a simple
algebraic curve of genus~1.
