Bridge the two approximation regimes for antiferromagnetic Ising models

Characterize how to bridge the gap between the parameter regime admitting an FPRAS established for antiferromagnetic Ising models on random regular bipartite graphs and the distinct non-uniqueness regime admitting an FPTAS on regular bipartite expanders.

Background

The paper establishes an FPRAS when the uniform external field satisfies a constant upper bound and the interaction parameters obey a weak-interaction condition of the form $\lambda(1-\beta)\lesssim \Delta^{-1/2}$. It contrasts this with the FPTAS of Geisler et al., which applies in a different non-uniqueness regime characterized by a lower-order condition involving $(\ln \Delta)^{3/2}$ and powers of $\Delta$. Because these parameter ranges do not coincide, the authors explicitly identify an unresolved gap between the available approximation results.