I will discuss the problem of determining the number of infinite-volume ground states in the Edwards-Anderson (nearest neighbor) spin glass model on $Z^D$ for $D \geq 2$. There are no complete results for this problem even in $D=2$. I will focus on this case and explain recent results which go some way toward proving that (with zero external field, so that ground states come in pairs, related by a global spin flip) there is only a single ground state pair (GSP).
Our solution applies not to the full plane $Z^2$, but to a half-plane. In addition, rather than showing that a.s. (with respect to the quenched random coupling realization J) there is a single GSP, we show that there is a natural joint distribution on $J$ and GSP's such that for $a.e$. $J$, the conditional distribution on GSP's given $J$ is supported on only a single GSP.
The methods used are a combination of percolation-like geometric arguments with translation invariance (in one of the two coordinate directions of the half-plane) and uses as a main tool the ``excitation metastate'' which is a probability measure on GSP's and on how they change as one or more individual couplings vary.
(Joint work with Louis-Pierre Arguin, Chuck Newman, and Dan Stein.)