Cutoff and mixing trichotomy for the simple random walk on random digraphs
Giacomo Passuello

April 30th 2025

Cutoff and mixing trichotomy for the simple random walk on random digraphs

Giacomo Passuello

In his talk Giacomo studies the mixing behaviour of random walks on the directed Chung-Lu graph and a directed version of the stochastic block model.

This talk will be broadcast at 13:30 GMT / 14:30 CET / 15:30 EET, April 30th, 2025 on MS Teams only.

Meeting ID: 393 527 089 878
Passcode: dsm7py

Abstract

We study the mixing behaviour of the simple random walk on two different random digraphs (directed graphs). We first consider the Chung-Lu digraph, which belongs to the family of inhomogeneous Erdős–Rényi digraphs, in a weakly dense regime where the random walk is irreducible. As the size n of the graph grows, the model exhibits with high probability a cutoff with a Gaussian window, namely an abrupt decay of the distance to equilibrium, at the threshold timescale log n/ log log n. We then introduce a digraph featuring a community structure, inspired by the stochastic block model. This second environment provides a mixing trichotomy, depending on the strength of connectivity among communities: we identify a subcritical regime, in which cutoff occurs; a supercritical regime, where the system has a sort of metastable behaviour; a critical regime, with mixed behaviour. We provide a characterization in terms of limit profiles, which enriches the analysis performed in the reversible setting.

Joint works with Alessandra Bianchi and Matteo Quattropani.

About Giacomo

Giacomo is currently a third year PhD student at the University of Padova under the supervision of Alessandra Bianchi. His research interests involve mixing times of random dynamics and random graphs.

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