Boundary Free Energies in Disordered Ising Models

The boundary correction to the free energy of a disordered Ising model depends on the boundary condition, the normalization, and the finite-volume sequence. We give a one-dimensional i.i.d. counterexample showing that the surface free-energy density need not be independent of the van Hove sequence and that the corresponding random correction need not converge in probability. For bounded couplings in the Dobrushin uniqueness regime on cubic boxes in dimensions $d\geq2$, we prove convergence of the normalized free-to-fixed boundary free-energy difference in expectation, almost surely, and in every $L^p$, $1\leq p<\infty$. For Gaussian boundary couplings, we derive an exact finite-volume interpolation identity and explain why it does not by itself imply a low-temperature surface limit. For a seam-flip free-energy difference $D_L$ across a set $S_L$ of independent symmetric bonds of variance $v$, we prove $\Var(D_L)\leq4v\abs{S_L}$; one-dimensional examples show that symmetry and finite moments alone do not determine a stiffness exponent, and the low-temperature Gaussian Edwards--Anderson problem remains open.