  We study ground states of Ising models with random ferromagnetic couplings,
proving the triviality of all zero-temperature metastates. This unexpected
result sheds a new light on the properties of these systems, putting strong
restrictions on their possible ground state structure. Open problems related to
existence of interface-supporting ground states are stated and an
interpretation of the main result in terms of first-passage and random surface
models in a random environment is presented.
