We give an 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 non-uniqueness 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