Localization on $4$ sites for Vertex-reinforced random walks on $\mathbb Z$.

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 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.

Mots clés: Reinforced random walk; Localization; Urn model

Domaines: Probabilités [math.PR]

Anne-Laure Basdevant, Bruno Schapira, Arvind Singh. Localization on $4$ sites for Vertex-reinforced random walks on $\mathbb Z$.. 2012. ⟨hal-00656106v2⟩