Characterization of cutoff for reversible Markov chains

• Riddhipratim Basu ^[1] ; Jonathan Hermon ^[2] ; Yuval Peres ^[3]
  1. [1] Stanford University

     Stanford University

     Estados Unidos

     
  2. [2] University of California, Berkeley

     University of California, Berkeley

     Estados Unidos

     
  3. [3] The Microsoft Research - University of Trento Centre for Computational and Systems Biology

     The Microsoft Research - University of Trento Centre for Computational and Systems Biology

     Trento, Italia

     

  Mostrar afiliaciones +
• Localización: Annals of probability: An official journal of the Institute of Mathematical Statistics, ISSN 0091-1798, Vol. 45, Nº. 3, 2017, págs. 1448-1487
• Idioma: inglés
• Enlaces
  • Texto completo
• Resumen

  • A sequence of Markov chains is said to exhibit (total variation) cutoff if the convergence to stationarity in total variation distance is abrupt. We consider reversible lazy chains. We prove a necessary and sufficient condition for the occurrence of the cutoff phenomena in terms of concentration of hitting time of “worst” (in some sense) sets of stationary measure at least αα, for some α∈(0,1)α∈(0,1).

    We also give general bounds on the total variation distance of a reversible chain at time tt in terms of the probability that some “worst” set of stationary measure at least αα was not hit by time tt. As an application of our techniques, we show that a sequence of lazy Markov chains on finite trees exhibits a cutoff iff the product of their spectral gaps and their (lazy) mixing-times tends to ∞∞.