corresponding variance $$\begin{aligned}
\label{eq:sigma}
\sigma^2 & := & \frac{1}{N}\sum_{v\in V}\deg(v)\left\{\log\left(\deg(v)-1\right)-\mu\right\}^2.\end{aligned}$$ We will also need some control on the third absolute moment: $$\begin{aligned}
\label{eq:varrho}
\varrho & := & \frac{1}{N}\sum_{v\in V}\deg(v)\left|\log\left(\deg(v)-1\right)-\mu\right|^3.\end{aligned}$$ It might help the reader to think of $\mu,\sigma$ and $\varrho$ as being fixed, or bounded away from $0$ and $\infty$. However, we only impose the following (much weaker) condition: $$\begin{aligned}
\label{assume:sigma}
{\frac{\sigma^2}{\mu^{3}}} >> \frac{(\log\log N)^2}{{\log N}} & \textrm{ and } & \frac{\sigma^3}{\varrho\sqrt{\mu}}>>\frac 1{\sqrt{{\log N}}}.\end{aligned}$$ Our main result states that on most graphs with degrees $(\deg(v))_{v\in V}$, the <span style="font-variant:small-caps;">nbrw</span> exhibits a remarkable behaviour, visible on Figure \[fig:cutoff\] and known as a *cutoff*: the distance to equilibrium remains close to $1$ for a rather long time, roughly $$\begin{aligned}
\label{eq:t_star}
t_\star:=\frac{\log N}{\mu},\end{aligned}$$ and then abruptly drops to nearly $0$ over a much shorter time scale[^1], of order $$\begin{aligned}
\label{eq:omega_star}
\omega_\star:=\sqrt{\frac{{\sigma^2\log N}}{\mu^{3}}}.\end{aligned}$$ Moreover, the cutoff shape inside this window approaches a surprisingly simple function $\Phi\colon{\ensuremath{\mathbb{R}}}\to[0,1]$, namely the tail distribution of the standard normal: $$\Phi(\lambda):=\frac{1}{2\pi}\int_{\lambda}^\infty e^{-\frac {u^2}2}\,{\rm d} u.$$ It is remarkable that this limit shape does not depend at all on the precise degrees.

\[thm:main\] For every $0<\varepsilon<1$, $$\begin{aligned}
\frac{t_{\textsc{mix}}(\varepsilon)-t_\star}{\omega_\star} & \xrightarrow[]{{\ensuremath{\mathbb{P}}}} & {\Phi}^{-1}(\varepsilon).\end{aligned}$$ Equivalently, for $t=t_\star+\lambda\omega_\star+o(w_\star)$ with $\lambda\in{\ensuremath{\mathbb{R}}}$ fixed, we have ${\ensuremath{\mathcal{D}}}(t)  