Tangential Surface Pressures and Boundary Localization in Disordered Edwards--Anderson Models

Abstract

We study boundary free energies in disordered Edwards--Anderson Ising models. For rectangular strips of fixed width, we prove that the expected free-to-fixed boundary correction, divided by the tangential length, converges for every finite inverse temperature and at zero temperature. The proof combines a uniform almost-additivity estimate with product-measure concentration, yielding self-averaging along tangential intervals. We derive an exact finite-volume Gaussian interpolation identity and prove that a uniform exponential boundary-mixing condition implies boundary localization with an explicit exponential rate. The tangential pressure theorem extends to all dimensions and symmetric coupling laws with finite first moment; under a product-concentration hypothesis, self-averaging is obtained along tangential cubes. Finally, a subcritical open-bond criterion, verified explicitly for sparse signed couplings, gives low-temperature localization without assuming an infinite-volume Gibbs-state property.