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The main contribution of the current study is two-fold. First, we investigatethe energy landscape of the Ising and Potts models on finite two-dimensionallattices without external fields in the low temperature regime. The completeanalysis of the energy landscape of these models was unknown because of itscomplicated plateau saddle structure between the ground states. We characterizethis structure completely in terms of a random walk on the set of sub-trees ofa ladder graph. Second, we provide a considerable simplification of thewell-known potential-theoretic approach to metastability. In particular, byreplacing the role of variational principles such as the Dirichlet and Thomsonprinciples with an $H^1$-approximation of the equilibrium potential, we developa new method that can be applied to non-reversible dynamics as well in a simplemanner. As an application of this method, we analyze metastable behavior of notonly the reversible Metropolis-Hastings dynamics, but also of severalinteresting non-reversible dynamics associated with the low-temperature Isingand Potts models explained above, and derive the Eyring-Kramers law and theMarkov chain model reduction of these models.

Approximation method to metastability: an application to non-reversible, two-dimensional Ising and Potts models without external fields