STUCK WALKS: A CONJECTURE OF ERSCHLER, TOTH AND WERNER

• 成果类型: Article
• 署名作者: Kious, Daniel
• 刊物名称: ANNALS OF PROBABILITY
• ISSN/ISSBN: 0091-1798
• DOI: 10.1214/14-AOP991
• 发表日期: 2016
• 页码: 883-923
• 关键词: reinforced-random-walk; localization

摘要

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, Toth and Werner proved in [Probab. Theory Related Fields 154 (2012) 149-163] that, for any L >= 1, if the parameter alpha belongs to a certain interval (alpha(L+1), alpha(L)), 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 alpha is an element of (1,+infinity) = (alpha(2), alpha(1)), then the walk localizes a.s. on 3 sites.