Comptes Rendus
no. 6
Probability theory
Vertex-reinforced random walk on Z with sub-square-root weights is recurrent

Presented by: Jean-François Le Gall

Jun Chen1; Gady Kozma2
1 Division of the Humanities and Social Sciences, California Institute of Technology, Pasadena, CA 91125, USA
2 Faculty of Mathematics and Computer Science, Weizmann Institute of Science, POB 26, 76100, Rehovot, Israel
Comptes Rendus. Mathématique, Volume 352 (2014) no. 6, pp. 521-524.

We prove that vertex-reinforced random walk on Z with weight of order kα, for α∈[0,1/2), is recurrent. This confirms a conjecture of Volkov for α<1/2. The conjecture for α∈[1/2,1) remains open.

On démontre que toute marche aléatoire renforcée par sommets sur Z avec poids de l'ordre de kα, pour α∈[0,1/2), est récurrente. Ce résultat confirme une conjecture de Volkov pour α<1/2. La conjecture reste ouverte pour α∈[1/2,1).

Received:
Accepted:
Published online:
DOI: 10.1016/j.crma.2014.03.019

Jun Chen 1; Gady Kozma 2

1 Division of the Humanities and Social Sciences, California Institute of Technology, Pasadena, CA 91125, USA
2 Faculty of Mathematics and Computer Science, Weizmann Institute of Science, POB 26, 76100, Rehovot, Israel 
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Jun Chen; Gady Kozma. Vertex-reinforced random walk on $ \mathbb{Z}$ with sub-square-root weights is recurrent. Comptes Rendus. Mathématique, Volume 352 (2014) no. 6, pp. 521-524. doi : 10.1016/j.crma.2014.03.019. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2014.03.019/

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