Independent sets of a given size and structure in the hypercube

TL;DR

This paper analyzes the asymptotic number and structural properties of large independent sets in high-dimensional hypercubes, extending previous results and developing new combinatorial tools.

Contribution

It extends the asymptotic enumeration of independent sets in hypercubes to all densities and introduces a multivariate local CLT for independent set structures.

Findings

Asymptotic formulas for independent sets of various sizes in hypercubes

A multivariate local CLT for structural features of independent sets

Development of combinatorial tools using polymer models and cluster expansion

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.