Critical percolation on random regular graphs

We show that for all d\in \{3,\ldots ,n-1\} the size of the largest component of a random d-regular graph on n vertices around the percolation threshold p=1/(d-1) is \Theta (n^{2/3}), with high probability. This extends known results for fixed d\geq 3 and for d=n-1, confirming a prediction of Nachmias and Peres on a question of Benjamini. As a corollary, for the largest component of the percolated random d-regular graph, we also determine the diameter and the mixing time of the lazy random walk. In contrast to previous approaches, our proof is based on a simple application of the switching method.