Spinodals in the Ising Model with Short-Ranged Interactions in the Hyperbolic Plane: True, Dynamic, and Thermodynamic*

In the Euclidean Plane, the short-ranged Ising model does not have true metastability -- the lifetime of the ``metastable'' state is always finite -- and therefore it has no true spinodal field. They do have, though, a ``thermodynamic spinodal'', in which the mode of decay switches from the nucleation and growth of one or more critical droplets to a subcritical fluctuation spanning the system and then growing in a quasi-one-dimensional fashion. They also have a ``dynamic spinodal'', which marks the transition from deterministic decay to stochastic decay. These two ``spinodals'' are distinct from each other and have different values. In contrast, the short-ranged Ising model in the hyperbolic plane \textbf{does} have true metastability, and, in the limit of large systems, its true spinodal, thermodynamic spinodal, and dynamic spinodal all coincide.