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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 $L\ge1$, if the parameter $α$ belongs to a certain interval $(α_{L+1},α_L)$, 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 $α\in(1,+\infty)=(α_2,α_1)$, then the walk localizes a.s. on $3$ sites.