Independent sets of a given size and structure in the hypercube

We determine the asymptotics of the number of independent sets of size $\lfloor β2^{d-1} \rfloor$ in the discrete hypercube $Q_d = \{0,1\}^d$ for any fixed $β\in [0,1]$ as $d \to \infty$, extending a result of Galvin for $β\in [1-1/\sqrt{2},1]$. 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 $λ>0$. 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.

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