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-Diaconis, 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.
Type de document :
Pré-publication, Document de travail
2015
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https://hal.archives-ouvertes.fr/hal-01187523
Contributeur : Justin Salez <>
Soumis le : jeudi 27 août 2015 - 09:37:07
Dernière modification le : lundi 29 mai 2017 - 14:23:10
Document(s) archivé(s) le : samedi 28 novembre 2015 - 10:22:57

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  • HAL Id : hal-01187523, version 1
  • ARXIV : 1508.06600

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Charles Bordenave, Pietro Caputo, Justin Salez. Random walk on sparse random digraphs. 2015. <hal-01187523>

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