  In this article, we investigate the energy landscape and metastable behavior
of the Ising and Potts models on two-dimensional square or hexagonal lattices
in the low temperature regime, especially in the absence of an external
magnetic field. The energy landscape of these models without an external field
is known to have a huge and complex saddle structure between ground states. In
the small volume regime where the lattice is finite and fixed, the
aforementioned complicated saddle structure has been successfully analyzed in
[20] for two or three dimensional square lattices when the inverse temperature
tends to infinity. In this article, we consider the large volume regime where
the size of the lattice grows to infinity. We first establish an asymptotically
sharp threshold such that the ground states are metastable if and only if the
inverse temperature is larger than the threshold in a suitable sense. Then, we
carry out a detailed analysis of the energy landscape and rigorously establish
the Eyring-Kramers formula when the inverse temperature is sufficiently larger
than the previously mentioned sharp threshold. The proof relies on detailed
characterization of dead-ends appearing in the vicinity of optimal transitions
between ground states and on combinatorial estimation of the number of
configurations lying on a certain energy level.
