Diaconis, Persi; Saloff-Coste, Laurent Comparison theorems for reversible Markov chains. (English) Zbl 0799.60058 Ann. Appl. Probab. 3, No. 3, 696-730 (1993). This paper develops methods for getting upper and lower bounds on the eigenvalues \[ 1=\beta_ 0>\beta_ 1\geq\cdots\geq\beta_{| X|- 1}\geq-1, \] of an irreducible reversible Markov matrix \(P\) by comparison with a second reversible chain on the same state space \(X\). It extends the ideas introduced by the authors [Ann. Probab. 21, No. 4, 2131-2156 (1993; Zbl 0790.60011)], where the random walks on finite groups were considered, and the geometric properties as the diameter and covering number of an associated graph, which appeared in [first author and D. Stroock, Ann. Appl. Probab. 1, No. 1, 36-61 (1991; Zbl 0731.60061)] are utilized. Apparently the bounds given in the paper have wide applications, especially in exclusion processes. Reviewer: W.-Z.Yang (Taipei) Cited in 4 ReviewsCited in 132 Documents MSC: 60J10 Markov chains (discrete-time Markov processes on discrete state spaces) 60F05 Central limit and other weak theorems Keywords:exclusion process; Poincaré inequalities; Bernoulli-Laplace diffusion; upper and lower bounds on the eigenvalues; irreducible reversible Markov matrix Citations:Zbl 0790.60011; Zbl 0731.60061 PDFBibTeX XMLCite \textit{P. Diaconis} and \textit{L. Saloff-Coste}, Ann. Appl. Probab. 3, No. 3, 696--730 (1993; Zbl 0799.60058) Full Text: DOI