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Long Term Behaviour of a Reversible System of Interacting Random Walks

Janson, Svante; Shcherbakov, Vadim and Volkov, Stanislav LU (2019) In Journal of Statistical Physics
Abstract

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... (More)

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.

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Please use this url to cite or link to this publication:
author
organization
publishing date
type
Contribution to journal
publication status
epub
subject
keywords
Lyapunov function, Markov chain, Martingale, Random walk, Recurrence, Renewal measure, Return time, Transience
in
Journal of Statistical Physics
publisher
Springer
external identifiers
  • scopus:85061183245
ISSN
0022-4715
DOI
10.1007/s10955-019-02244-0
language
English
LU publication?
yes
id
9ccb3200-9bdc-4a26-bc64-e1ec65a2a232
date added to LUP
2019-02-22 12:50:28
date last changed
2019-03-12 04:23:11
@article{9ccb3200-9bdc-4a26-bc64-e1ec65a2a232,
  abstract     = {<p>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.</p>},
  author       = {Janson, Svante and Shcherbakov, Vadim and Volkov, Stanislav},
  issn         = {0022-4715},
  keyword      = {Lyapunov function,Markov chain,Martingale,Random walk,Recurrence,Renewal measure,Return time,Transience},
  language     = {eng},
  publisher    = {Springer},
  series       = {Journal of Statistical Physics},
  title        = {Long Term Behaviour of a Reversible System of Interacting Random Walks},
  url          = {http://dx.doi.org/10.1007/s10955-019-02244-0},
  year         = {2019},
}