Cutoff for random walk on random graphs with a community structure

We consider a variant of the configuration model with an embedded community structure and study the mixing properties of a simple random walk on it. Every vertex has an internal $\mathrm{deg}^{\text{int}}\geq 3$ and an outgoing $\mathrm{deg}^{\text{out}}$ number of half-edges. Given a stochastic matrix $Q$, we pick a random perfect matching of the half-edges subject to the constraint that each vertex $v$ has $\mathrm{deg}^{\text{int}}(v)$ neighbours inside its community and the proportion of outgoing half-edges from community $i$ matched to a half-edge from community $j$ is $Q(i,j)$. Assuming the number of communities is constant and they all have comparable sizes, we prove the following dichotomy: simple random walk on the resulting graph exhibits cutoff if and only if the product of the Cheeger constant of $Q$ times $\log n$ (where $n$ is the number of vertices) diverges. In [4], Ben-Hamou established a dichotomy for cutoff for a non-backtracking random walk on a similar random graph model with 2 communities. We prove the same characterisation of cutoff holds for simple random walk.