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arXiv · 1208.6375

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

Michel Benaïm·Olivier Raimond·Bruno Schapira
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Abstract

We study Vertex-Reinforced-Random-Walk on the 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.

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Michel Benaïm, Olivier Raimond, Bruno Schapira. 2013-10-07. Strongly Vertex-Reinforced-Random-Walk on the complete graph. https://arxiv.org/abs/1208.6375

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