Published online by Cambridge University Press: 06 January 2022
We determine the asymptotics of the number of independent sets of size $\lfloor \beta 2^{d-1} \rfloor$ in the discrete hypercube
$Q_d = \{0,1\}^d$ for any fixed
$\beta \in (0,1)$ as
$d \to \infty$, extending a result of Galvin for
$\beta \in (1-1/\sqrt{2},1)$. Moreover, we prove a multivariate local central limit theorem for structural features of independent sets in
$Q_d$ drawn according to the hard-core model at any fixed fugacity
$\lambda>0$. In proving these results we develop several general tools for performing combinatorial enumeration using polymer models and the cluster expansion from statistical physics along with local central limit theorems.