Mixing cutoff for simple random walks on the Chung-Lu digraph

In this paper, we are interested in the mixing behaviour of simple random walks on inhomogeneous directed graphs. We focus our study on the Chung-Lu digraph, which is an inhomogeneous network that generalizes the Erd\H{o}s-R\'enyi digraph. In particular, under the Chung-Lu model, edges are included in the graph independently and according to given Bernoulli laws, so that the average degrees are fixed. To guarantee the a.s. existence of a unique reversible measure, which is implied by the strong connectivity of the graph, we assume that the average degree grows logarithmically in the size $n$ of the graph. In this weakly dense regime, we prove that the total variation distance to equilibrium displays a cutoff behaviour at the entropic time of order $\log(n)/\log\log(n)$. Moreover, we prove that on a precise window, the cutoff profile converges to the Gaussian tail function. This is qualitatively similar to what was proved in [6,7,8] for the directed configuration model, where degrees are deterministically fixed. In terms of statistical ensembles, our analysis provides an extension of these cutoff results from a hard to a soft-constrained model.