Resumen de Stuck walks: A conjecture of Erschler, Tóth and Werner

Daniel Kious

• 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.