The component structure of dense random subgraphs of the hypercube

TL;DR

This paper analyzes the structure of the fragment in random subgraphs of hypercubes when edges are present with probability less than 0.5, providing detailed asymptotic estimates and distribution descriptions of components.

Contribution

It extends previous work by giving detailed asymptotic estimates and joint distribution descriptions of component sizes in the hypercube fragment for p<0.5.

Findings

Asymptotic estimates for mean number of components of each size

Distribution and joint distribution of component sizes

Characterization of the fragment structure in hypercube subgraphs

Abstract

Given $p \in (0,1)$, we let $Q_p= Q_p^d$ be the random subgraph of the $d$-dimensional hypercube $Q^d$ where edges are present independently with probability $p$. It is well known that, as $d \rightarrow \infty$, if $p>\frac12$ then with high probability $Q_p$ is connected; and if $p<\frac12$ then with high probability $Q_p$ consists of one giant component together with many smaller components which form the `fragment'. Here we fix $p \in (0,\frac12)$, and investigate the fragment, and how it sits inside the hypercube. In particular we give asymptotic estimates for the mean numbers of components in the fragment of each size, and describe their asymptotic distributions and indeed their joint distribution, much extending earlier work of Weber.