Algorithms for #BIS-hard problems on expander graphs

We give a fully polynomial-time approximation scheme (FPTAS) and an efficient sampling algorithm for the high-fugacity hard-core model on bounded-degree bipartite expander graphs and the low-temperature ferromagnetic Potts model on bounded-degree expander graphs. The results apply, for example, to random (bipartite) Δ-regular graphs, for which no efficient algorithms were known for these problems (with the exception of the Ising model) in the nonuniqueness regime of the infinite Δ-regular tree. We also find efficient counting and sampling algorithms for proper 𝑞-colorings of random Δ-regular bipartite graphs when 𝑞 is sufficiently small as a function of Δ.