Localization of a vertex reinforced random walks on $\Z$ with sub-linear weights
Résumé
We consider a vertex reinforced random walk on the integer lattice with sub-linear reinforcement. Under some assumptions on the regular variation of the weight function, we characterize whether the walk gets stuck on a finite interval. When this happens, we estimate the size of the localization set. In particular, we show that, for any odd number $N$ larger than or equal to $5$, there exists a vertex reinforced random walk which localizes with positive probability on exactly $N$ consecutive sites.
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