Localization on 4 sites for vertex-reinforced random walks on Z

We characterize nondecreasing weight functions for which the associated one-dimensional vertex reinforced random walk (VRRW) localizes on 4 sites. A phase transition appears for weights of order nloglogn: for weights growing faster than this rate, the VRRW localizes almost surely on, at most, 4 sites, whereas for weights growing slower, the VRRW cannot localize on less than 5 sites. When w is of order nloglogn, the VRRW localizes almost surely on either 4 or 5 sites, both events happening with positive probability.