Zero-Temperature Fluctuations in Short-Range Spin Glasses

We consider the energy difference restricted to a finite volume for certain pairs of incongruent ground states (if they exist) in the d-dimensional Edwards–Anderson Ising spin glass at zero temperature. We prove that the variance of this quantity with respect to the couplings grows proportionally to the volume in any d≥ 2. An essential aspect of our result is the use of the excitation metastate. As an illustration of potential applications, we use this result to restrict the possible structure of spin glass ground states in two dimensions.