Annulus crossings and partner distances in two-dimensional Edwards–Anderson ground states

We study the set U of unsatisfied dual edges in ground states of the two-dimensional Edwards–Anderson spin glass with i.i.d. couplings J, assuming only ℙ(J = 0) = 0 and 𝔼|J| < ∞, so that the ±J model is included. For every centre x and r ≥ 8 there is an event of probability at least 1 − 9e^(−r/4), depending only on the couplings within distance 2r of x, on which no ground state has more than K edge-disjoint unsatisfied paths crossing the annulus between radii r and 2r; this holds on the same event for infinite-volume ground states, for ground states of finite systems containing a neighbourhood of the annulus, under free, fixed, periodic or antiperiodic boundary conditions, and for configurations in the support of zero-temperature metastates, and K depends only on the law of |J|. Consequently, for a ground state and a decomposition of U into trails pairing up the frustrated plaquettes whose joint law with J is translation invariant, the probability that the origin is frustrated and its trail has radius at least R is at most (64K + 6500)R⁻²; a density version holds without invariance, and U has at most K ends. The proof closes up crossings along two low-weight squares and applies minimality to the enclosed region, a deterministic, local form of the region flips of Tessler (at zero temperature) and of Berger and Tessler (at low temperature). We also prove that U has only finite components in translation-invariant ground states for all such laws; Tessler proved this for symmetric continuous couplings with full support, and Berger and Tessler for Gaussian couplings; we remove symmetry, continuity and full support at the cost of the moment condition 𝔼|J| < ∞. Finally, every finite simple path in U is a geodesic for the weights |J|, the infinite-volume form of a classical fact on minimum-weight joins.