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.
@article{arxiv.1807.04804,
title = {Algorithms for #BIS-hard problems on expander graphs},
author = {Matthew Jenssen and Peter Keevash and Will Perkins},
journal= {arXiv preprint arXiv:1807.04804},
year = {2020}
}