# Independent sets in the hypercube revisited

## TL;DR

This paper improves the asymptotic understanding of independent sets in the hypercube by combining combinatorial and statistical physics methods, providing sharper results and detailed structural insights.

## Contribution

It introduces a novel combination of graph container methods with cluster expansion techniques to refine asymptotics and structural descriptions of independent sets in the hypercube.

## Key findings

- Sharper asymptotic formulas for independent sets
- Detailed probabilistic structure of typical independent sets
- Answers to open questions posed by Galvin

## Abstract

We revisit Sapozhenko's classic proof on the asymptotics of the number of independent sets in the discrete hypercube $\{0,1\}^d$ and Galvin's follow-up work on weighted independent sets. We combine Sapozhenko's graph container methods with the cluster expansion and abstract polymer models, two tools from statistical physics, to obtain considerably sharper asymptotics and detailed probabilistic information about the typical structure of (weighted) independent sets in the hypercube. These results refine those of Korshunov and Sapozhenko and Galvin, and answer several questions of Galvin.