A simple proof of almost percolation on G(n;p)

TL;DR

This paper provides a simplified proof for the near-percolation phenomenon in bootstrap percolation on random graphs, improving probability bounds with elementary martingale and giant component arguments.

Contribution

It offers a straightforward proof technique that strengthens existing probability bounds for bootstrap percolation on G(n,p).

Findings

Enhanced probability bounds for infected vertices.

Simplified proof approach using martingales.

Improved understanding of percolation thresholds.

Abstract

We consider bootstrap percolation on the binomial random graph $G(n,p)$ with infection threshold $r\in \mathbb{N}$, an infection process which starts from a set of initially infected vertices and in each step every vertex with at least $r$ infected neighbours becomes infected. We improve the results of Janson, \L uczak, Turova, and Valier (2012) by strengthening the probability bounds on the number of infected vertices at the end of the process, using simple arguments based on martingales and giant components.