no. 6
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, Jun1 ; Kozma, Gady2
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

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).

Reçu le :
Accepté le :
Publié le :
DOI : 10.1016/j.crma.2014.03.019

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 
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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

[1] Davis, Burgess Reinforced random walk, Probab. Theory Relat. Fields, Volume 84 (1990) no. 2, pp. 203-229 springer.com (Available at:)

[2] Gideon Amir, Itai Benjamini, Ori Gurel-Gurevich, Gady Kozma, Random walk in changing environment, unpublished manuscript, circa 2006.

[3] Pemantle, Robin Vertex-reinforced random walk, Probab. Theory Relat. Fields, Volume 92 (1992) no. 1, pp. 117-136 springer.com upenn.edu/~pemantle (Available at:)

[4] Pemantle, Robin; Volkov, Stanislav Vertex-reinforced random walk on Z has finite range, Ann. Probab., Volume 27 (1999) no. 3, pp. 1368-1388 projecteuclid.org (Available at:)

[5] Schapira, Bruno A 0–1 law for vertex-reinforced random walks on Z with weight of order kα, α<1/2, Electron. Commun. Probab., Volume 17 (2012) no. 22 ejpecp.org (Available at:)

[6] Singh, Arvind Recurrence for vertex-reinforced random walks on Z with weak reinforcements, 2014 (Available at:) | arXiv

[7] Tarrès, Pierre Vertex-reinforced random walk on Z eventually gets stuck on five points, Ann. Probab., Volume 32 (2004) no. 3B, pp. 2650-2701 projecteuclid.org (Available at:)

[8] Volkov, Stanislav Phase transition in vertex-reinforced random walks on Z with non-linear reinforcement, J. Theor. Probab., Volume 19 (2006) no. 3, pp. 691-700 springer.com lth.se/.../s.volkov (Available at:)

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