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January 2000 On the speed of random walks on graphs
Bálint Virág
Ann. Probab. 28(1): 379-394 (January 2000). DOI: 10.1214/aop/1019160123


Lyons, Pemantle and Peres asked whether the asymptoticlower speed in an infinite tree is bounded by the asymptoticspeed in the regular tree with the same average number of branches. In the more general setting of random walks on graphs, we establish a bound on the expected value of the exit time from a vertex set in terms of the size and distance from the origin of its boundary, and prove this conjecture. We give sharp bounds for limiting speed (or, when applicable,sublinear rate of escape) in terms of growth properties of the graph. For trees, we get a bound for the speed in terms of the Hausdorff dimension of the harmonicmeasure on the boundary. As a consequence, two conjectures of Lyons, Pemantle and Peres are resolved, and a new bound is given for the dimension of the harmonicmeasure defined by the biased random walk on a Galton–Watson tree.


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Bálint Virág. "On the speed of random walks on graphs." Ann. Probab. 28 (1) 379 - 394, January 2000.


Published: January 2000
First available in Project Euclid: 18 April 2002

zbMATH: 1044.60034
MathSciNet: MR1756009
Digital Object Identifier: 10.1214/aop/1019160123

Primary: 60F15 , 60J15
Secondary: 60J80

Keywords: graph , hitting time , Random walk , Rate of escape , Speed

Rights: Copyright © 2000 Institute of Mathematical Statistics


Vol.28 • No. 1 • January 2000
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