Random walk on sparse random digraphs

Abstract

A finite ergodic Markov chain exhibits cutoff if its distance to equilibrium remains close to its initial value over a certain number of iterations and then abruptly drops to near 0 on a much shorter time scale. Originally discovered in the context of card shuffling (Aldous and Diaconis in Am Math Mon 93:333–348, 1986), this remarkable phenomenon is now rigorously established for many reversible chains. Here we consider the non-reversible case of random walks on sparse directed graphs, for which even the equilibrium measure is far from being understood. We work under the configuration model, allowing both the in-degrees and the out-degrees to be freely specified. We establish the cutoff phenomenon, determine its precise window and prove that the cutoff profile approaches a universal shape. We also provide a detailed description of the equilibrium measure.

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Fig. 1

Notes

  1. 1.

    Using an arbitrary deterministic ordering of the set of all tails to break ties.

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Correspondence to Charles Bordenave.

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Bordenave, C., Caputo, P. & Salez, J. Random walk on sparse random digraphs. Probab. Theory Relat. Fields 170, 933–960 (2018). https://doi.org/10.1007/s00440-017-0796-7

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Mathematics Subject Classification

  • Primary 05C80
  • 05C81