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arXiv · 2106.09709

Independent sets of a given size and structure in the hypercube

Matthew Jenssen·Will Perkins·Aditya Potukuchi
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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.

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Matthew Jenssen, Will Perkins, Aditya Potukuchi. 2021-06-17. Independent sets of a given size and structure in the hypercube. https://arxiv.org/abs/2106.09709

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