Localization of a vertex reinforced random walk on mathbb{Z } with sub-linear weight

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.