Long Term Behaviour of a Reversible System of Interacting Random Walks

Research output: Contribution to journalArticle


This paper studies the long-term behaviour of a system of interacting random walks labelled by vertices of a finite graph. We show that the system undergoes phase transitions, with different behaviour in various regions, depending on model parameters and properties of the underlying graph. We provide the complete classification of the long-term behaviour of the corresponding continuous time Markov chain, identifying whether it is null recurrent, positive recurrent, or transient. The proofs are partially based on the reversibility of the model, which allows us to use the method of electric networks. We also provide some alternative proofs (based on the Lyapunov function method and the renewal theory), which are of interest in their own right, since they do not require reversibility and can be applied to more general situations.


External organisations
  • Uppsala University
  • Royal Holloway University of London
Research areas and keywords

Subject classification (UKÄ) – MANDATORY

  • Transport Systems and Logistics


  • Lyapunov function, Markov chain, Martingale, Random walk, Recurrence, Renewal measure, Return time, Transience
Original languageEnglish
Pages (from-to)71-96
JournalJournal of Statistical Physics
Issue number1
Early online date2019
Publication statusPublished - 2019
Publication categoryResearch