Monte Carlo Studies of the Isoperimetric Dimension of Growing Droplets in Metastable Decay of the Ising Model on Small-World Graphs

For the Ising model on a regular, nearest-neighbor lattice of less than 6 dimensions, metastable decay occurs via the nucleation of critical droplets; subcritical droplets are biased toward shrinkage, whereas supercritical droplets are biased toward growth. The size of a critical droplet is governed by the competition between the coupling of the magnetic field to the volume of the droplet, which lowers the free energy, and the coupling of the droplet of the stable state to metastable state at the boundary of the droplet, which increases the free energy. This competition between volume effects and surface effects makes the isoperimetric dimension relevant to metastable decay. The simulations discussed here are for a triangular lattice with a small percentage of ``small-world'' connections. The system initially has only one ``down'' spin, from which the droplet grows; switching is irreversible and only occurs for ``up'' spins adjacent to at least one ``down'' spin.