Expansion in supercritical random subgraphs of the hypercube and its consequences

TL;DR

This paper investigates the properties of the giant component in supercritical random subgraphs of the hypercube, revealing bounds on expansion, diameter, and mixing time, and implications for graph circumference and Hadwiger number.

Contribution

It provides new bounds on the expansion, diameter, and mixing time of the giant component in supercritical hypercube subgraphs, answering open questions and extending previous results.

Findings

Vertex-expansion of the giant component is inverse polynomial in d.

Diameter and mixing time are polynomial in d.

Lower bounds on circumference and Hadwiger number are established.

Abstract

It is well-known that the behaviour of a random subgraph of a $d$-dimensional hypercube, where we include each edge independently with probability $p$, undergoes a phase transition when $p$ is around $\frac{1}{d}$. More precisely, standard arguments show that just below this value of $p$ all components of this graph have order $O(d)$ with probability tending to one as $d \to \infty$ (whp for short), whereas Ajtai, Koml\'{o}s and Szemer\'{e}di [Largest random component of a $k$-cube, Combinatorica 2 (1982), no. 1, 1--7; MR0671140] showed that just above this value, in the supercritical regime, whp there is a unique `giant' component of order $\Theta\left(2^d\right)$. We show that whp the vertex-expansion of the giant component is inverse polynomial in $d$. As a consequence we obtain polynomial in $d$ bounds on the diameter of the giant component and the mixing time of the lazy random walk on the giant component, answering questions of Bollob\'{a}s, Kohayakawa and {\L}uczak [On the diameter and radius of random subgraphs of the cube, Random Structures and Algorithms 5 (1994), no. 5, 627--648; MR1300592] and of Pete [A note on percolation on $\mathbb{Z}^d$: isoperimetric profile via exponential cluster repulsion, Electron. Commun. Probab. 13 (2008), 377--392; MR2415145]. Furthermore, our results imply lower bounds on the circumference and Hadwiger number of a random subgraph of the hypercube in this regime of $p$ which are tight up to polynomial factors in $d$.