Stuck walks: A conjecture of Erschler, Tóth and Werner

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 . belongs to a certain interval (.L+1,.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 ..(1,+.)=(.2,.1), then the walk localizes a.s. on 3 sites.