The sharp threshold for jigsaw percolation in random graphs

TL;DR

This paper establishes the precise sharp threshold for jigsaw percolation in random graphs, showing it occurs at a specific probability product of 1/(4n ln n), refining previous phase transition results.

Contribution

The paper proves the exact sharp threshold for jigsaw percolation in random graphs, improving understanding of the phase transition point.

Findings

Threshold occurs at 1/(4n ln n)

Sharp phase transition established

Refines previous probabilistic bounds

Abstract

We analyse the jigsaw percolation process, which may be seen as a measure of whether two graphs on the same vertex set are `jointly connected'. Bollob\'as, Riordan, Slivken and Smith proved that when the two graphs are independent binomial random graphs, whether the jigsaw process percolates undergoes a phase transition when the product of the two probabilities is $\Theta\left( \frac{1}{n\ln n} \right)$. We show that this threshold is sharp, and that it lies at $\frac{1}{4n\ln n}$.