The zero-temperature Edwards–Anderson model on strips and cylinders

We study ground states of the Edwards–Anderson spin glass on strips and cylinders ℤ × Λ, where Λ is a finite connected graph and the couplings are i.i.d. If the coupling law is atomless and either charges (0, ∞) or Λ is bipartite, or if it is atomless with ess sup|J| / ess inf|J| > |E(Λ)| + |Λ| + 1, then almost surely the strip has exactly one ground-state pair. Its satisfaction field is a finitary factor of the couplings, with an exponential tail on the coding radius in the first case, and finite-volume ground states with free, periodic or disorder-independent end conditions converge to it in satisfaction. For laws with a bounded density that is bounded below by a positive constant near 0, the probability that the satisfaction of a horizontal edge at the centre of a box of half-length L depends on the end condition lies between c/L and C/L; thus the order 1/L of Chatterjee's lower bound for this boundary sensitivity, proved for boxes in dimension at least two, is sharp on strips. For symmetric atomless laws, Chatterjee's argument run on a ring gives the lower bound 1/(384L). In the first case the bond overlap tends to one as the perturbation of the couplings vanishes, so the strip analogue of Chatterjee's question on bond chaos in the stronger sense has a negative answer. For laws with a density as above, the site overlap with free ends is chaotic under resampling exactly when pL → ∞; under the Gaussian Ornstein–Uhlenbeck perturbation it is chaotic as soon as pL² → ∞. In the first case, and if the couplings have finite variance, the ground-state energy satisfies a central limit theorem with positive variance. So does the zero-temperature free-to-fixed face correction of Wang, Zhu and Chueng; this is the zero-temperature case of a central limit theorem they list as a next step. The proofs reduce the strip to a one-dimensional chain: blocks on which all couplings lie in a short window (under the ratio condition, a single column with a dominant spanning tree) fix the configuration on a column, up to sign, in every ground state, and between two such forced columns the ground state is decided by a single effective coupling.