  Let $d\ge 3$ be a fixed integer, $p\in (0,1)$, and let $n\geq 1$ be a
positive integer such that $dn$ is even. Let $\mathbb{G}(n, d, p)$ 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 $|\mathcal{C}_{\text{max}}|$ be the
size of the largest component in $\mathbb{G}(n, d, p)$. We show that, when $p$
is of the form $p=(d-1)^{-1}(1+\lambda n^{-1/3})$ for $\lambda\in \mathbb{R}$,
and $A$ is large, \begin{align*}
\mathbb{P}(|\mathcal{C}_{\text{max}}|>An^{2/3})\asymp
A^{-3/2}e^{-\frac{A^3(d-1)(d-2)}{8d^2}+\frac{\lambda
A^2(d-2)^2}{2d(d-1)}-\frac{\lambda^2 A(d-1)}{2(d-2)}}. \end{align*} 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 $An^{2/3}$.
