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September 2007 Attracting edge and strongly edge reinforced walks
Vlada Limic, Pierre Tarrès
Ann. Probab. 35(5): 1783-1806 (September 2007). DOI: 10.1214/009117906000001097

Abstract

The goal is to show that an edge-reinforced random walk on a graph of bounded degree, with reinforcement weight function W taken from a general class of reciprocally summable reinforcement weight functions, traverses a random attracting edge at all large times.

The statement of the main theorem is very close to settling a conjecture of Sellke [Technical Report 94-26 (1994) Purdue Univ.]. An important corollary of this main result says that if W is reciprocally summable and nondecreasing, the attracting edge exists on any graph of bounded degree, with probability 1. Another corollary is the main theorem of Limic [Ann. Probab. 31 (2003) 1615–1654], where the class of weights was restricted to reciprocally summable powers.

The proof uses martingale and other techniques developed by the authors in separate studies of edge- and vertex-reinforced walks [Ann. Probab. 31 (2003) 1615–1654, Ann. Probab. 32 (2004) 2650–2701] and of nonconvergence properties of stochastic algorithms toward unstable equilibrium points of the associated deterministic dynamics [C. R. Acad. Sci. Sér. I Math. 330 (2000) 125–130].

Citation

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Vlada Limic. Pierre Tarrès. "Attracting edge and strongly edge reinforced walks." Ann. Probab. 35 (5) 1783 - 1806, September 2007. https://doi.org/10.1214/009117906000001097

Information

Published: September 2007
First available in Project Euclid: 5 September 2007

zbMATH: 1131.60036
MathSciNet: MR2349575
Digital Object Identifier: 10.1214/009117906000001097

Subjects:
Primary: 60G50
Secondary: 60J10 , 60K35

Keywords: Attracting edge , martingale , reinforced walk

Rights: Copyright © 2007 Institute of Mathematical Statistics

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Vol.35 • No. 5 • September 2007
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