Independent sets of a given size and structure in the hypercube

• DOI: 10.1017/s0963548321000559
• Publication Date: 2022-01-06T09:24:05Z
• Authors: Matthew Jenssen; Will Perkins; Aditya Potukuchi
• Keywords: Central limit theorem; Enumeration

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