Stuck walks: A conjecture of Erschler, T\'oth 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, T\'{o}th and Werner proved in [Probab. Theory Related Fields 154 (2012) 149-163] that, for any $L\ge1$, 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 $\alpha\in(1,+\infty)=(\alpha_2,\alpha_1)$, then the walk localizes a.s. on $3$ sites.