Spinodal Field and Surface Free Energy of the Ising Model on the \{5,4\} Tiling of the Hyperbolic Plane*

Consider the ferromagnetic Ising model on a two-dimensional lattice, with all the spins initially \textit{up} but with a weak \textit{down} magnetic field, evolving under a single-spin-flip Metropolis dynamic. If the lattice lies in the Euclidean plane -- for example, if it is the square lattice --- a droplet of \textit{down} spins (appearing as a thermal excitation) can decrease the free energy of the system by growing if it is larger than a finite critical size. In the hyperbolic plane, however, beneath a \textbf{spinodal field} $H_{sp}$ it is impossible to nucleate a critical droplet. Monte Carlo simulations for finite regions of the \{5,4\} tiling with mean-field boundary conditions show that $H_{sp}^{2/3}$ is approximately a linear function of temperature, which should be expected at least in the neighborhood of the critical temperature. Assuming that the droplets are circular, a first estimate of the surface free energy can be made.