Localization on 4 sites for Vertex-reinforced random walks on $\mathbb   Z$
  We characterize non-decreasing 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 $n\log \log n$: 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 $n\log \log n$, the VRRW localizes almost surely on either 4 or 5 sites, both events happening with positive probability. 
Anne-Laure Basdevant (MODAL'X), Bruno Schapira (LM-Orsay), Arvind   Singh (LM-Orsay)
math.PR