Anderson-like localization transition of random walks with resetting

Denis Boyer, Andrea Falcón-Cortés, Luca Giuggioli, Satya N Majumdar

Research output: Contribution to journalArticle (Academic Journal)peer-review

16 Citations (Scopus)
105 Downloads (Pure)


We study several lattice random walk models with stochastic resetting to previously visited sites which exhibit a phase transition between an anomalous diffusive regime and a localization regime where diffusion is suppressed. The localized phase settles above a critical resetting rate, or rate of memory use, and the probability density asymptotically adopts in this regime a non-equilibrium steady state similar to that of the well known problem of diffusion with resetting to the origin. The transition occurs because of the presence of a single impurity site where the resetting rate is lower than on other sites, and around which the walker spontaneously localizes. Near criticality, the localization length diverges with a critical exponent that falls in the same class as the self-consistent theory of Anderson localization of waves in random media. The critical dimensions are also the same in both problems. Our study provides analytically tractable examples of localization transitions in path-dependent, reinforced stochastic processes, which can be also useful for understanding spatial learning by living organisms.
Original languageEnglish
Article number 053204
Number of pages28
JournalJournal of Statistical Mechanics: Theory and Experiment
Publication statusPublished - 23 May 2019

Structured keywords

  • Engineering Mathematics Research Group


  • Classical phase transitions
  • Diusion
  • Stochastic processes


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