Independent sets of a given size and structure in the hypercube

Abstract

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.

Typo corrected in Section 3.2.1