CRITICAL PERCOLATION ON RANDOM REGULAR GRAPHS

We show that for all d epsilon {3,..., 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 >= 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.