Some results on periodic–antiperiodic domain walls in Edwards–Anderson ground states

We collect results on the relative domain wall between the periodic and the antiperiodic ground states of the Edwards–Anderson spin glass. In the first part we work on the d-dimensional torus of side L with i.i.d. couplings whose law μ is symmetric, has no atoms and has a finite Poincaré constant C_μ. The difference D_L of the two ground-state energies satisfies Var D_L ≤ 4C_μ ρ_L L^(d−1), where ρ_L ≥ 1/L is the probability that a fixed edge in the twisted direction lies on the wall. This sharpens, by the factor ρ_L, the bound Var D_L ≤ 4Var(μ) L^(d−1) of Wang, Zhu and Chueng, which rests on the same antisymmetry, at the price of the Poincaré constant C_μ ≥ Var(μ) (equal for Gaussian couplings), which cannot be replaced by Var(μ). Consequently, if Var D_L ≥ cL^(d−1) along a subsequence, a translation-invariant joint limit of the two ground states charges pairs that are incongruent in the sense of Newman and Stein; in contrapositive form, if in every such limit a fixed edge in the twisted direction lies on the wall with probability zero, then Var D_L = o(L^(d−1)). For Gaussian couplings we bound the Ornstein–Uhlenbeck correlations of the gradient of D_L, a signed field carried by the wall, and we prove a positive-temperature version of the variance bound. In the second part we work in the half-plane ℤ × ℤ₊, where Arguin, Damron, Newman and Stein proved that periodic side boundary conditions select a single ground-state pair. We show that the same pair is selected by antiperiodic sides and by constant fixed tops, and, after averaging over horizontal translates, by free sides and arbitrary fixed tops; these are half-plane versions of remarks of Newman and Stein for the full plane. As a corollary, the periodic–antiperiodic wall in their boxes crosses every row an odd number of times, yet, at every fixed height, it leaves each fixed window with probability tending to one and meets the row in o(n) edges in expectation.