Mixing times for a constrained lsing process on the torus at low density

• Natesh S. Pillai ^[1] ; Aaron Smith ^[2]
  1. [1] Harvard University

     Harvard University

     City of Cambridge, Estados Unidos

     
  2. [2] University of Ottawa

     University of Ottawa

     Canadá

     
• Localización: Annals of probability: An official journal of the Institute of Mathematical Statistics, ISSN 0091-1798, Vol. 45, Nº. 2, 2017, págs. 1003-1070
• Idioma: inglés
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• Resumen

  • We study a kinetically constrained Ising process (KCIP) associated with a graph GG and density parameter pp; this process is an interacting particle system with state space, the location of the particles. The number of particles at stationarity follows the Binomial(|G|,pBinomial⁡(|G|,p) distribution, conditioned on having at least one particle. The “constraint” in the name of the process refers to the rule that a vertex cannot change its state unless it has at least one neighbour in state “1”. The KCIP has been proposed by statistical physicists as a model for the glass transition, and more recently as a simple algorithm for data storage in computer networks. In this note, we study the mixing time of this process on the torus in the low-density regime p=c|G|p=c|G| for arbitrary; this regime is the subject of a conjecture of Aldous and is natural in the context of computer networks. Our results provide a counterexample to Aldous’ conjecture, suggest a natural modification of the conjecture, and show that this modification is correct up to logarithmic factors. The methods developed in this paper also provide a strategy for tackling Aldous’ conjecture for other graphs.