ABSTRACT We establish a refined version of a graph container lemma due to Galvin and discuss several applications related to the hard‐core model on bipartite expander graphs. Given a graph and , the hard‐core model on at activity is the probability distribution on independent sets in given by . As one of our main applications, we show that the hard‐core model at activity on the hypercube exhibits a ‘structured phase’ for in the following sense: in a typical sample from , most vertices are contained in one side of the bipartition of . This improves upon a result of Galvin, which establishes the same for . As another application, we establish a fully polynomial‐time approximation scheme (FPTAS) for the hard‐core model on a ‐regular bipartite ‐expander, with fixed, when . This improves upon the bound due to the first author, Perkins and Potukuchi. We discuss similar improvements to results of Galvin‐Tetali, Balogh‐Garcia‐Li and Kronenberg‐Spinka.
Jenssen et al. (Thu,) studied this question.