The sharp threshold for jigsaw percolation in random graphs

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. Bollobas, 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 $Thetaleft( frac{1}{nln n} right)$. We show that this threshold is sharp, and that it lies at $frac{1}{4nln n}$.