Independent sets in random subgraphs of the hypercube

TL;DR

This paper investigates the asymptotic number of independent sets in random subgraphs of the hypercube across various edge retention probabilities, extending previous work on the hypercube's independent sets and the hardcore model.

Contribution

It generalizes prior results by analyzing the independent sets in $Q_{d,p}$ for a wide range of $p$, including sparse, constant, and near-complete regimes, using the antiferromagnetic Ising model.

Findings

Asymptotic enumeration of independent sets for various $p$ regimes.

Extension of results to the hardcore model on $Q_{d,p}$.

Connections established between the hardcore model and the antiferromagnetic Ising model.

Abstract

Let $Q_{d,p}$ be the random subgraph of the $d$-dimensional hypercube $\{0,1\}^d$, where each edge is retained independently with probability $p$. We study the asymptotic number of independent sets in $Q_{d,p}$ as $d \to \infty$ for a wide range of parameters $p$, including values of $p$ tending to zero as fast as $\frac{C\log d}{d^{1/3}}$, constant values of $p$, and values of $p$ tending to one. The results extend to the hardcore model on $Q_{d,p}$, and are obtained by studying the closely related antiferromagnetic Ising model on the hypercube, which can be viewed as a positive-temperature hardcore model on the hypercube. These results generalize previous results by Galvin, Jenssen and Perkins on the hard-core model on the hypercube, corresponding to the case $p=1$, which extended Korshunov and Sapozhenko's classical result on the asymptotic number of independent sets in the hypercube.