paper

Attracting edge and strongly edge reinforced walks

arXiv:math/0604200 · 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 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 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].

Published in at http://dx.doi.org/10.1214/009117906000001097 the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org)

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