Open Access
October, 1995 Eigenvalues of Random Walks on Groups
Richard Stong
Ann. Probab. 23(4): 1961-1981 (October, 1995). DOI: 10.1214/aop/1176987811


In this paper we discuss and apply a novel method for bounding the eigenvalues of a random walk on a group $G$ (or equivalently on its Cayley graph). This method works by looking at the action of an Abelian normal subgroup $H$ of $G$ on $G$. We may then choose eigenvectors which fall into representations of $H$. One is then left with a large number (one for each representation of $H$) of easier problems to analyze. This analysis is carried out by new geometric methods. This method allows us to give bounds on the second largest eigenvalue of random walks on nilpotent groups with low class number. The method also lets us treat certain very easy solvable groups and to give better bounds for certain nice nilpotent groups with large class number. For example, we will give sharp bounds for two natural random walks on groups of upper triangular matrices.


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Richard Stong. "Eigenvalues of Random Walks on Groups." Ann. Probab. 23 (4) 1961 - 1981, October, 1995.


Published: October, 1995
First available in Project Euclid: 19 April 2007

zbMATH: 0852.60078
MathSciNet: MR1379176
Digital Object Identifier: 10.1214/aop/1176987811

Primary: 60J10

Keywords: Eigenvalues , nilpotent groups , Random walks

Rights: Copyright © 1995 Institute of Mathematical Statistics

Vol.23 • No. 4 • October, 1995
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