  We consider short-range classical spin glasses, or other disordered systems,
consisting of Ising spins. For a low-temperature Gibbs state in infinite size
in such a system, for given random bonds, it is controversial whether its
decomposition into pure states will be trivial or non-trivial. We undertake a
general study of the overall structure of this problem, based on metastates,
which are essential to prove the existence of a thermodynamic limit. A
metastate is a probability distribution on Gibbs states, for given disorder,
that satisfies certain covariance properties. First, we prove that any
metastate can be decomposed as a mixture of indecomposable metastates, and that
all Gibbs states drawn from an indecomposable metastate are alike
macroscopically. Next, we consider stochastic stability of a metastate under
random perturbations of the disorder, and prove that any metastate is
stochastically stable. Using related methods and older results, we prove that
if the pure-state decomposition of any Gibbs states drawn from an
indecomposable metastate is countably infinite, then the weights in the
decomposition follow a Poisson-Dirichlet distribution with a fixed value of the
single parameter describing such distributions, and also that if the overlap
takes a finite number of values, then the pure states are organized as an
ultrametric space, as in $k$-RSB. Dynamically-frozen states play a role in the
analysis of Gibbs states drawn from a metastate, either as states or as parts
of states. Using a mapping into real Hilbert space, we prove further results
about Gibbs states, and classify them into six types. Metastate-average states
are studied, and can be related to states arising dynamically at long times
after a quench from high temperature, under some conditions.
