Random walks on dynamic configuration models: a trichotomy

We consider a dynamic random graph on $n$ vertices that is obtained by starting from a random graph generated according to the configuration model with a prescribed degree sequence and at each unit of time randomly rewiring a fraction ${\alpha }_n$ of the edges. We are interested in the mixing time of a random walk without backtracking on this dynamic random graph in the limit as $n\to \mathrm{\infty }$, when ${\alpha }_n$ is chosen such that $\lim_{n\to\infty} \alpha_n (\log n)^2 = \beta \in [0,\infty]$. In [1] we found that, under mild regularity conditions on the degree sequence, the mixing time is of order $1\mathrm{/}\sqrt{{\alpha }_n}$ when $\beta =\mathrm{\infty }$. In the present paper we investigate what happens when $\beta \in [0,\mathrm{\infty })$. It turns out that the mixing time is of order $\log n$, with the scaled mixing time exhibiting a one-sided cutoff when $\beta \in (0,\mathrm{\infty })$ and a two-sided cutoff when $\beta =0$. The occurrence of a one-sided cutoff is a rare phenomenon. In our setting it comes from a competition between the time scales of mixing on the static graph, as identified by Ben-Hamou and Salez [4], and the regeneration time of first stepping across a rewired edge.