the observation that if $({\mathcal{P}}_1^\star,{\mathcal{C}}_1^\star)$ and $({\mathcal{P}}_2^\star,{\mathcal{C}}_2^\star)$ both satisfy conditions (1)–(3), then so does $({\mathcal{P}}_1^\star\cup{\mathcal{P}}_2^\star,{\mathcal{C}}_1^\star
\cup{\mathcal{C}}_2^\star)$. Note that $$\begin{array}{lll}
&{\mathcal{H}}(\xi)<\Gamma^\star+{\mathcal{H}}(\boxminus) &\forall\,\xi \in {\mathcal{P}}^\star,\\[0.1cm]
&{\mathcal{H}}(\xi)=\Gamma^\star+{\mathcal{H}}(\boxminus) &\forall\,\xi \in {\mathcal{C}}^\star.
\end{array}$$

It is shown in Bovier and den Hollander [@BdH15 Chapter 16] that *subject to hypothesis* (H) the following three theorems hold.

\[thm:critdrop\] $\lim_{\beta\to\infty} P^{G,\beta}_\boxminus(\tau_{{\mathcal{C}}^\star}<\tau_\boxplus
\mid \tau_\boxplus < \tau_\boxminus)=1$.

\[thm:nucltime\] There exists a $K^\star \in (0,\infty)$ such that $$\lim_{\beta\to\infty}
e^{-\beta\Gamma^\star}\,E^{G,\beta}_\boxminus(\tau_\boxplus) = K^\star.$$

\[thm:explaw\] (a) $\lim_{\beta\to\infty} \lambda^{G,\beta}_\beta\,E^{G,\beta}_\boxminus(\tau_\boxplus)=1$.\
(b) $\lim_{\beta\to\infty} P^{G,\beta}_\boxminus(\tau_\boxplus/
E^{G,\beta}_\boxminus(\tau_\boxplus)>t) = e^{-t}$ for all $t \geq 0$.

The proofs of Theorems \[thm:critdrop\]–\[thm:explaw\] in [@BdH15] do not rely on the details of the graph $G$, provided it is finite, connected and non-oriented (i.e., allowed moves are possible in both directions). For concrete choices of $G$, the task is to verify hypothesis (H) and to identify the triple (see Fig. \[fig-abstractdoublewell\]) $$\label{eq:triple}
\big({\mathcal{C}}^\star,\Gamma^\star,K^\star\big).$$ For lattice graphs this task has been carried out successfully (even for several classes of dynamics: see [@BdH15 Chapters 17–18]). For random graphs, however, the triplet in is random, and describing it represents a *very serious challenge*. In what follows we focus on a particular class of random graphs called the *Configuration Model*. But before doing so, we first summarise what is 