  Let $G=G(d)$ be a random graph with a given degree sequence $d$, such as a
random $r$-regular graph where $r\ge 3$ is fixed and $n=|G|\to\infty$. We study
the percolation phase transition on such graphs $G$, i.e., the emergence as $p$
increases of a unique giant component in the random subgraph $G[p]$ obtained by
keeping edges independently with probability $p$. More generally, we study the
emergence of a giant component in $G(d)$ itself as $d$ varies. We show that a
single method can be used to prove very precise results below, inside and above
the `scaling window' of the phase transition, matching many of the known
results for the much simpler model $G(n,p)$. This method is a natural extension
of that used by Bollobas and the author to study $G(n,p)$, itself based on work
of Aldous and of Nachmias and Peres; the calculations are significantly more
involved in the present setting.
