Vertex-reinforced random walk on $ \mathbb{Z}$ with sub-square-root weights is recurrent

Chen, Jun; Kozma, Gady

doi:10.1016/j.crma.2014.03.019

Probability theory

Vertex-reinforced random walk on

Z

with sub-square-root weights is recurrent
[Récurrence d'une marche aléatoire renforcée par sommets sur

Z

avec poids inférieur à racine carrée]

Présenté par : Jean-François Le Gall

Chen, Jun ¹ ; Kozma, Gady 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

Comptes Rendus. Mathématique, Tome 352 (2014) no. 6, pp. 521-524.

Résumé
Abstract

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

We prove that vertex-reinforced random walk on $Z$ with weight of order $k^{α}$ , for $α \in [0, 1 / 2)$ , is recurrent. This confirms a conjecture of Volkov for $α < 1 / 2$ . The conjecture for $α \in [1 / 2, 1)$ remains open.

Reçu le : 2014-02-04
Accepté le : 2014-03-07
Publié le : 2014-05-13

DOI : 10.1016/j.crma.2014.03.019

Affiliations des auteurs :

Chen, Jun ¹ ; Kozma, Gady ²

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     author = {Chen, Jun and Kozma, Gady},
     title = {Vertex-reinforced random walk on $ \mathbb{Z}$ with sub-square-root weights is recurrent},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {521--524},
     publisher = {Elsevier},
     volume = {352},
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Chen, Jun; Kozma, Gady. Vertex-reinforced random walk on $ \mathbb{Z}$ with sub-square-root weights is recurrent. Comptes Rendus. Mathématique, Tome 352 (2014) no. 6, pp. 521-524. doi : 10.1016/j.crma.2014.03.019. http://www.numdam.org/articles/10.1016/j.crma.2014.03.019/

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