Subextensive Random Boundary Perturbations in the Short-Range Edwards--Anderson Model

We consider the nearest-neighbor Edwards--Anderson Ising model on cubic boxes with random perturbations supported at the boundary.

We prove that any perturbation admitting an energy envelope that is negligible compared with the volume, both in expectation and almost surely, leaves the limiting quenched specific free energy unchanged.

For independent finite-variance bulk disorder in dimension at least two, an Efron--Stein estimate also yields almost-sure self-averaging along the full sequence of boxes.

The hypotheses are verified for i.i.d. scalar surface fields, fixed random exterior spins, and periodic wrap-around bonds, with an $ O(L^{-1}) $ comparison of quenched means.

The result concerns the specific free energy and does not assert convergence of finite-volume Gibbs measures.