Metastable states in spin glasses and disordered ferromagnets

We study analytically M-spin-flip stable states in disordered short-ranged Ising models (spin glasses and ferromagnets) in all dimensions and for all M. Our approach is primarily dynamical, and is based on the convergence of ${\sigma }^t,$ a zero-temperature dynamical process with flips of lattice animals up to size M and starting from a deep quench, to a metastable limit ${\sigma }^{\infty }.$ The results (rigorous and nonrigorous, in infinite and finite volumes) concern many aspects of metastable states: their numbers, basins of attraction, energy densities, overlaps, remanent magnetizations, and relations to thermodynamic states. For example, we show that their overlap distribution is a $\delta$ function at zero. We also define a dynamics for $M=\infty ,$ which provides a potential tool for investigating ground state structure.