  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.
