Localization of a vertex reinforced random walks on $\Z$ with sub-linear   weights
  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. 
Anne-Laure Basdevant (MODAL'X), Bruno Schapira (LM-Orsay), Arvind   Singh (LM-Orsay)
math.PR