Strongly Vertex-Reinforced-Random-Walk on a complete graph
Résumé
We study Vertex-Reinforced-Random-Walk (VRRW) on a complete graph with weights of the form w(n) = n α , with α > 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 α 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.
Domaines
Probabilités [math.PR]
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