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Comparing eigenvalue bounds for Markov chains: When does Poincaré beat Cheeger? (English) Zbl 0935.60057

From the authors’ summary: “The Poincaré and Cheeger bounds are useful for the second largest eigenvalue of a reversible Markov chain. There are versions of these bounds which involve choosing paths. This paper studies these path-related bounds and shows that Poincaré bound is superior to the Cheeger bound for simple random walk on a finite group with any symmetric generating set.”
For related papers see P. Diaconis and D. Stroock [Ann. Appl. Probab. 1, No. 1, 36-61 (1991; Zbl 0731.60061)], M. Jerrum and A. Sinclair [SIAM J. Comput. 18, No. 6, 1149-1178 (1989; Zbl 0723.05107)] and A. Sinclair [Comb. Probab. Comput. 1, No. 4, 351-370 (1992; Zbl 0801.90039)].

MSC:

60J10 Markov chains (discrete-time Markov processes on discrete state spaces)
60C05 Combinatorial probability
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