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

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

Abstract : We study Vertex-Reinforced-Random-Walk on the complete graph with weights of the form $wn=n^\alpha$, 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. At the end of the paper, we also prove a general result ensuring that any strongly reinforced VRRW on any bounded degree graph localizes a.s. on a finite subgraph.

Keywords : Vertex-Reinforced-Random-Walk complete graph stochastic approximation

Autor: Michel Benaïm - Olivier Raimond - Bruno Schapira -

Fuente: https://hal.archives-ouvertes.fr/

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