BEHAVIOR OF DROPLETS FOR A CLASS OF GLAUBER DYNAMICS AT VERY LOW-TEMPERATURE

We consider a class of Glauber dynamics for the two-dimensional nearest neighbor ferromagnetic Ising model in which the rate with which each spin flips depends only on the increment in energy caused by its flip in a monotonic non-increasing fashion. We extend to this class results previously shown for the particular case of Metropolis dynamics [NS]. We show that for fixed volume and external field 0 < h < 1, at very low temperature small rectangular droplets of spins +1 in a sea of spins -1 tend to shrink, while large droplets tend to grow and cover the whole system. The threshold between the two behaviors is sharply defined, the critical length being 2/h. An example is given which shows that without the assumption of monotonicity of the rates this result may be false. We use the result on critical droplets to show that starting from the configuration with all spins down, the systems in the class that we consider evolve in a metastable fashion until the configuration with all spins up is reached. For similar systems in higher dimensions we show that under analogous conditions on the rates, small enough droplets are likely to shrink, while large enough droplets are likely to grow.