Open Access
January 1997 Vertex-reinforced random walks and a conjecture of Pemantle
Michel Benaïm
Ann. Probab. 25(1): 361-392 (January 1997). DOI: 10.1214/aop/1024404292


We discuss and disprove a conjecture of Pemantle concerning vertex-reinforced random walks.

The setting is a general theory of non-Markovian discrete-time random 4 processes on a finite space $E = {1, \dots, d}$, for which the transition probabilities at each step are influenced by the proportion of times each state has been visited. It is shown that, under mild conditions, the asymptotic behavior of the empirical occupation measure of the process is precisely related to the asymptotic behavior of some deterministic dynamical system induced by a vector field on the $d - 1$ unit simplex. In particular, any minimal attractor of this vector field has a positive probability to be the limit set of the sequence of empirical occupation measures. These properties are used to disprove a conjecture and to extend some results due to Pemantle. Some applications to edge-reinforced random walks are also considered.


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Michel Benaïm. "Vertex-reinforced random walks and a conjecture of Pemantle." Ann. Probab. 25 (1) 361 - 392, January 1997.


Published: January 1997
First available in Project Euclid: 18 June 2002

zbMATH: 0873.60044
MathSciNet: MR1428513
Digital Object Identifier: 10.1214/aop/1024404292

Primary: 60J10
Secondary: 34C35 , 34F05

Keywords: attractors , chain recurrence , random perturbations of dynamical systems , reinforced random walks

Rights: Copyright © 1997 Institute of Mathematical Statistics

Vol.25 • No. 1 • January 1997
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