Independent sets of a given size and structure in the hypercube

We determine the asymptotics of the number of independent sets of size ⌊β2 d−1 ⌋ in the discrete hypercube Qd = {0, 1} d for any fixed β ∈ [0, 1] as d → ∞, extending a result of Galvin for β ∈ [1−1/ √ 2, 1]. Moreover, we prove a multivariate local central limit theorem for structural features of independent sets in Qd drawn according to the hard core model at any fixed fugacity λ > 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.