Localization in lattice and continuum models of reinforced random walks

✍ Scribed by K.J. Painter; D. Horstmann; H.G. Othmer

• Publisher: Elsevier Science
• Year: 2003
• Tongue: English
• Weight: 408 KB
• Volume: 16
• Category: Article
• ISSN: 0893-9659
• DOI: 10.1016/s0893-9659(03)80060-5

✦ Synopsis

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 wlues 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.