Structural analysis of Gibbs states and metastates in short-range classical spin glasses: indecomposable metastates, dynamically-frozen states, and metasymmetry

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.