  An important but little-studied property of spin glasses is the stability of
their ground states to changes in one or a finite number of couplings. It was
shown in earlier work that, if multiple ground states are assumed to exist,
then fluctuations in their energy differences --- and therefore the possibility
of multiple ground states --- are closely related to the stability of their
ground states. Here we examine the stability of ground states in two models,
one of which is presumed to have a ground state structure that is qualitatively
similar to other realistic short-range spin glasses in finite dimensions.
