Stuck Walks: A Conjecture Of Erschler, Toth 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, 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.