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

In this article, we are interested in the mixing behaviour of simple random walks on inhomogeneous directed graphs. We focus our study on Chung–Lu digraphs, which are inhomogeneous networks that generalize Erdős-Rényi digraphs, and where edges are included independently and according to given Bernoulli laws. digraph. To guarantee the a.s. existence of a unique equilibrium measure, we assume that the average degree grows logarithmically in the size n of the graph. In this weakly sparse regime, we prove that the total variation distance to equilibrium displays a cutoff behaviour at the entropic time of order log(n)/loglog(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 by Bordenave et al. and Cai et al. for the directed configuration model. In terms of statistical ensembles, our analysis provides an extension of these cutoff results from a hard to a soft-constrained model.