The phase transition in the configuration model

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 $n=\mathrm{\mid }G\mathrm{\mid }\to \mathrm{\infty }$. 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.