Localization of a vertex reinforced random walks on $\Z$ with sub-linear weights

Anne-Laure Basdevant (1) , Bruno Schapira (2) , Arvind Singh (2)

1 MODAL'X - Modélisation aléatoire de Paris X
2 LM-Orsay - Laboratoire de Mathématiques d'Orsay

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.

Mots clés: Self-interacting random walk; regular variation; reinforcement; regular variation.

Domaines: Probabilités [math.PR]

Anne-Laure Basdevant, Bruno Schapira, Arvind Singh. Localization of a vertex reinforced random walks on $\Z$ with sub-linear weights. Probability Theory and Related Fields, 2014, ⟨10.1007/s00440-013-0502-3⟩. ⟨hal-00716115v2⟩