Communications in Mathematical Physics

, Volume 358, Issue 3, pp 1151–1185 | Cite as

Non-fixation for Conservative Stochastic Dynamics on the Line

  • Riddhipratim BasuEmail author
  • Shirshendu Ganguly
  • Christopher Hoffman


We consider activated random walk (ARW), a model which generalizes the stochastic sandpile, one of the canonical examples of self organized criticality. Informally ARW is a particle system on \({\mathbb{Z}}\) with mass conservation. One starts with a mass density \({\mu > 0}\) of initially active particles, each of which performs a symmetric random walk at rate one and falls asleep at rate \({\lambda > 0}\). Sleepy particles become active on coming in contact with other active particles. We investigate the question of fixation/non-fixation of the process and show for small enough \({\lambda}\) the critical mass density for fixation is strictly less than one. Moreover, the critical density goes to zero as \({\lambda}\) tends to zero. This settles a long standing open question.


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Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2017

Authors and Affiliations

  • Riddhipratim Basu
    • 1
    Email author
  • Shirshendu Ganguly
    • 2
  • Christopher Hoffman
    • 3
  1. 1.International Centre for Theoretical SciencesTata Institute of Fundamental ResearchBangaloreIndia
  2. 2.Departments of Statistics and MathematicsUC BerkeleyBerkeleyUSA
  3. 3.Department of MathematicsUniversity of WashingtonSeattleUSA

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