Are there incongruent ground states in 2D Edwards-Anderson spin glasses?

We present a detailed proof of a previously announced result [1] supporting the absence of multiple (incongruent) ground state pairs for 2D Edwards-An derson spin glasses (with zero external field and, e.g., Gaussian couplings ): if two ground state pairs (chosen from metastates with, e.g., periodic b oundary conditions) on Z(2) are distinct, then the dual bonds where they di ffer form a single doubly-infinite, positive-density domain wall. It is an open problem to prove that such a situation cannot occur (or else to show m uch less likely in our opinion - that it indeed does happen) in these model s. Our proof involves an analysis of how (infinite-volume) ground states ch ange as (finitely many) couplings vary, which leads us to a notion of zero- temperature excitation metastates, that may be of independent interest.