A Refined Graph Container Lemma and Applications to the Hard‐Core Model on Bipartite Expanders

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.