On the critical fugacity of the hard-core model on regular bipartite graphs

We establish long-range order for the hard-core model on a finite, regular bipartite graph above a threshold fugacity given in terms of expansion parameters of the graph. The result applies to the $d$-dimensional hypercube graph and, more generally, to $d$-dimensional discrete tori of fixed side length, proving long-range order at fugacities $\lambda \ge \mathrm{\Omega }(\frac{\log d}{d})$. Furthermore, we use reflection positivity to transfer the result to the lattice ${\mathbb{Z}}^d$, verifying the long-standing belief that its critical fugacity is of the form $d^{-1+o(1)}$ as $d\to \mathrm{\infty }$.