The probability of unusually large components for critical percolation on random d-regular graphs

Let d > 3 be a fixed integer, p e (0; 1), and let n > 1 be a positive integer such that dn is even. Let G(n; d; p) be a (random) graph on n vertices obtained by drawing uniformly at random a d-regular (simple) graph on [n] and then performing independent p-bond percolation on it, i.e. we independently retain each edge with probability p and delete it with probability 1 - p. Let |C _max| be the size of the largest component in G(n; d; p). We show that, when p is of the form p = (d - 1)-1(1 + -n ^-1/3) for e R, and A is large, This improves on a result of Nachmias and Peres. We also give an analogous asymptotic for the probability that a particular vertex is in a component of size larger than An ^2/3.