We show that for all d in {3,…,n-1} the size of the largest component of a random d-regular graph on n vertices at the percolation threshold p=1/(d-1) is of order n^(2/3), with high probability. This extends known results for fixed d and for d=n-1, confirming a prediction of Nachmias and Peres on a question of Benjamini. In contrast to previous approaches, our proof is based on a simple application of the switching method. This is joint work with Felix Joos.

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