2024/02/23 by Alessandra Bianchi, Bianchi, Alessandra, Giacomo Passuello +1
Biochemistry, Genetics and Molecular Biology · Computer Science · Mathematics · #05C80 #05C81 #60J10 #Algorithms and Data Compression #DNA and Biological Computing #FOS: Mathematics #Probability (math.PR) #Stochastic processes and statistical mechanics
paper · pdf · doi:10.48550/arxiv.2402.15356
openalex publication_date 2024/02/23 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/01
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 Erdős-Rényi 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)/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 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.