Duality and free energy analyticity bounds for few-body Ising models with extensive homology rank

We consider pairs of few-body Ising models such that each model can be obtained from the dual of the other after freezing k spins on large-degree sites. Such a pair of Ising models can be interpreted as a two-chain complex with k being the rank of the first homology group. Our focus is on the case where k is extensive. In the presence of bond disorder, we prove the existence of a low-T weak-disorder region where additional summation over the defects has no effect on the free energy density f(T) in the thermodynamical limit, and of a high-T region where in the ferromagnetic case an extensive homological defect does not affect f(T). We also discuss the convergence of the high- and low-temperature series for the free energy density, prove the analyticity of limiting f(T) at high and low temperatures, and construct inequalities for the critical point(s) where analyticity is lost. As an application, we prove multiplicity of the conventionally defined critical points for Ising models on all {f, d} tilings of the infinite hyperbolic plane. For these infinite graphs, we show that critical temperatures with free and wired boundary conditions differ, T^(f)_c < T^(w)_c .