Localization in lattice and continuum models of reinforced random walks

K. J. Painter*, D. Horstmann, Hans G. Othmer

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

20 Citations (Scopus)


We study the singular limit of a class of reinforced random walks on a lattice for which a complete analysis of the existence and stability of solutions is possible. We show that at a sufficiently high total density, the global minimizer of a lattice 'energy' or Lyapunov functional corresponds to aggregation at one site. At lower values of the density the stable localized solution coexists with a stable spatially-uniform solution. Similar results apply in the continuum limit, where the singular limit leads to a nonlinear diffusion equation. Numerical simulations of the lattice walk show a complicated coarsening process leading to the final aggregation. © 2003 Elsevier Science Ltd. All rights reserved.

Original languageEnglish
Pages (from-to)375-381
Number of pages7
JournalApplied Mathematics Letters
Issue number3
Publication statusPublished - Apr 2003


  • Aggregation
  • Coarsening process
  • Forward-backward parabolic
  • Lattice walks

ASJC Scopus subject areas

  • Applied Mathematics


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