a given threshold $0<\varepsilon<1$ is known as the *mixing time*: $$\begin{aligned}
t_{\textsc{mix}}(\varepsilon) & := & \inf\left\{t\in{\ensuremath{\mathbb{N}}}\colon {\ensuremath{\mathcal{D}}}(t)<\varepsilon\right\}.\end{aligned}$$

Result.
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The present paper is concerned with the typical profile of the function $t\mapsto {\ensuremath{\mathcal{D}}}(t)$ under the so-called *configuration model*, i.e. when the pairing $\pi$ is chosen uniformly at random among the $(N-1)!!$ possible pairings of ${\ensuremath{\mathcal{X}}}$. In order to study large-size asymptotics, we let the vertex set $V$ and degree function $\deg\colon V\to{\ensuremath{\mathbb{N}}}$ depend on an implicit parameter $n\in{\ensuremath{\mathbb{N}}}$, which we omit from the notation for convenience. The same convention applies to all related quantities, such as $N$ or ${\ensuremath{\mathcal{X}}}$. All asymptotic statements are understood as $n\to\infty$. Our interest is in the *sparse regime*, where the number $N$ of half-edges diverges at a much faster rate than the maximum degree. Specifically, we assume that $$\begin{aligned}
\label{assume:Delta}
\Delta :=\max_{v\in V}\deg(v) & = & N^{o(1)}.\end{aligned}$$ As the behaviour of the <span style="font-variant:small-caps;">nbrw</span> at degree-2 vertices is deterministic, we will also assume, without much loss of generality, that $$\begin{aligned}
\label{assume:delta}
\min_{v\in V}\deg(v)&\geq& 3.\end{aligned}$$ Remarkably enough, the asymptotics in this regime depend on the degrees through two simple statistics: the mean logarithmic degree of an half-edge $$\begin{aligned}
\label{eq:mu}
\mu &:=& \frac{1}{N}\sum_{v\in V}\deg(v)\log\left(\deg(v)-1\right),  \end{aligned}$$ and the 