Dynamics of droplets in random Ising magnetic systems

We calculate the spin autocorrelation functions in random Ising magnets, within the framework of a Fokker-Planck model for the evolution of the droplet size. This model, which is equivalent to a diffusion process in a one-dimensional disordered medium, is treated by means of an effective-medium-type theory. Our analysis predicts a power-law decay for the autocorrelations in random-bond ferromagnets, of the form ${\mathit{t}}^{\mathrm{-}\mathit{Y}}$, where the nonuniversal exponent Y depends on both the temperature and the properties of the random-bond distribution. For weakly frustrated random-bond systems, with a finite fraction of antiferromagnetic defect bonds, the variation of Y with the parameters of the system exhibits a nonanalytic behavior at a freezing-type transition. We argue, however, that there are no corresponding nonanalyticities in the thermodynamic properties of the system. Spin autocorrelations in Ising spin glasses are also considered, by means of the effective-medium theory and the zero-temperature scaling hypothesis of Bray and Moore, Huse, and Fisher, and McMillan for the droplet energies. Our results are in agreement with the form of decay of autocorrelations proposed by Huse and Fisher.