Probability Theory and Related Fields

, Volume 159, Issue 1–2, pp 75–115 | Cite as

Localization of a vertex reinforced random walk on \(\mathbb{Z }\) with sub-linear weight

  • Anne-Laure Basdevant
  • Bruno Schapira
  • Arvind Singh


We consider a vertex reinforced random walk on the integer lattice with sub-linear reinforcement. Under some assumptions on the regular variation of the weight function, we characterize whether the walk gets stuck on a finite interval. When this happens, we estimate the size of the localization set. In particular, we show that, for any odd number \(N\) larger than or equal to \(5\), there exists a vertex reinforced random walk which localizes with positive probability on exactly \(N\) consecutive sites.


Self-interacting random walk Reinforcement Regular variation 

Mathematics Subject Classification

60K35 60J17 60J20 


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Copyright information

© Springer-Verlag Berlin Heidelberg 2013

Authors and Affiliations

  • Anne-Laure Basdevant
    • 1
  • Bruno Schapira
    • 2
  • Arvind Singh
    • 2
  1. 1.Laboratoire Modal’XUniversité Paris OuestParisFrance
  2. 2.Département de MathématiquesUniversité Paris XIParisFrance

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