Metastates and Replica Symmetry Breaking

In this review we define and discuss metastates, mathematical tools with general applicability to thermodynamic systems which are particularly useful when working with disordered or inhomogeneous short-range systems. In an infinite such system there may be many competing thermodynamic states, which can lead to the absence of a straightforward thermodynamic limit of local correlation functions. A metastate is a probability measure on the infinite-volume thermodynamic states that restores the connection between those states and the Gibbs states observed in finite volumes. After introducing the basic metastates and discussing their properties, we present possible scenarios for the spin-glass phase and discuss what the metastate approach reveals about how replica symmetry breaking would manifest itself in finite-dimensional short-range spin glasses.