Mathematics – Probability
Scientific paper
2006-04-09
Annals of Probability 2007, Vol. 35, No. 5, 1783-1806
Mathematics
Probability
Published in at http://dx.doi.org/10.1214/009117906000001097 the Annals of Probability (http://www.imstat.org/aop/) by the Ins
Scientific paper
10.1214/009117906000001097
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. S\'{e}r. I Math. 330 (2000) 125--130].
Limic Vlada
Tarres Pierre
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