\xrightarrow[]{{\ensuremath{\mathbb{P}}}}  \Phi(\lambda).$

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It is interesting to compare this with the $d-$regular case (i.e., $\deg\colon V\to{\ensuremath{\mathbb{N}}}$ constant equal to $d$) studied by @lubetzky2010cutoff: by a remarkably precise path counting argument, they establish cutoff within constantly many steps around $t_\star=\log N/\log(d-1)$. To appreciate the effect of heterogeneous degrees, recall that $\mu$ and $\sigma$ are the mean and variance of $\log D$, where $D$ is the degree of a uniformly sampled half-edge. Now, by Jensen’s Inequality, $$\begin{aligned}
t_\star & \geq & \frac{\log N}{\log{\ensuremath{\mathbb{E}}}[D]}\, ,\end{aligned}$$ and the less concentrated $D$, the larger the gap. The right-hand side is a well-known characteristic length in $G$, namely the typical inter-point distance (see e.g., [@remco]). One notable effect of heterogeneous degrees is thus that the mixing time becomes significantly larger than the natural graph distance. A heuristic explanation is as follows: in the regular case, all paths of length $t$ between two points are equally likely for the <span style="font-variant:small-caps;">nbrw</span>, and mixing occurs as soon as $t$ is large enough for many such paths to exist. In the non-regular case however, different paths have very different weights, and most of them actually have a negligible chance of being seen by the walk. Consequently, one has 