Supercritical Site Percolation on Regular Graphs

TL;DR

This paper studies vertex percolation on regular graphs, establishing conditions for the emergence of a giant component in the supercritical regime, with applications to hypercubes and pseudo-random graphs.

Contribution

It provides tight conditions for the giant component emergence in supercritical site percolation on regular graphs, resolving open questions for hypercubes and pseudo-random graphs.

Findings

Giant component appears at p=(1+ε)/(d-1) in supercritical regime

Results apply to hypercubes and pseudo-random graphs

Discusses differences between bond and site percolation

Abstract

We consider site (vertex) percolation on $d$-regular graphs, for both constant-degree and growing-degree cases. We give sufficient, and relatively tight, conditions for the emergence of the ``Erd\H{o}s-R\'enyi component phenomenon" in the supercritical regime $p=\frac{1+\epsilon}{d-1}$: namely, the appearance of a unique giant component of order $n/d$ in the percolated subgraph, with all other components being of size $O(\log n)$. Our main results apply both to the $d$-dimensional hypercube and to pseudo-random graphs, and resolve two open questions in these cases. We further discuss differences (and similarities) between bond (edge) percolation setting and site percolation setting.