Stuck walks: A conjecture of Erschler, Tóth and Werner

• Kious, Daniel ^[1]
  1. [1] Ecole Polytechnique Fédérale de Lausanne
• Localización: Annals of probability: An official journal of the Institute of Mathematical Statistics, ISSN 0091-1798, Vol. 44, Nº. 2, 2016, págs. 883-923
• Idioma: inglés
• DOI: 10.1214/14-AOP991
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• Resumen

  • In this paper, we work on a class of self-interacting nearest neighbor random walks, introduced in [Probab. Theory Related Fields 154 (2012) 149–163], for which there is competition between repulsion of neighboring edges and attraction of next-to-neighboring edges. Erschler, Tóth and Werner proved in [Probab. Theory Related Fields 154 (2012) 149–163] that, for any L≥1, if the parameter α belongs to a certain interval (αL+1,αL), then such random walks localize on L+2 sites with positive probability. They also conjectured that this is the almost sure behavior. We prove this conjecture partially, stating that the walk localizes on L+2 or L+3 sites almost surely, under the same assumptions. We also prove that, if α∈(1,+∞)=(α2,α1), then the walk localizes a.s. on 3 sites.