Strongly reinforced Vertex-Reinforced-Random-Walk on the complete graph

Michel Benaïm (1) , Olivier Raimond (2) , Bruno Schapira (3)

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

We study Vertex-Reinforced-Random-Walk on the complete graph with weights of the form $w(n)=n^\alpha$, with $\alpha>1$. Unlike for the Edge-Reinforced-Random-Walk, which in this case localizes a.s. on $2$ sites, here we observe various phase transitions, and in particular localization on arbitrary large sets is possible, provided $\alpha$ is close enough to $1$. Our proof relies on stochastic approximation techniques.

Mots clés: Vertex-Reinforced-Random-Walk; complete graph; stochastic approximation

Domaines: Probabilités [math.PR]

Michel Benaïm, Olivier Raimond, Bruno Schapira. Strongly reinforced Vertex-Reinforced-Random-Walk on the complete graph. 2012. ⟨hal-00724639v1⟩