Abstract
The Poincaré and Cheeger bounds are two useful bounds for the second largest eigenvalue of a reversible Markov chain. Diaconis and Stroock and Jerrum and Sinclair develop versions of these bounds which involve choosing paths. This paper studies these path-related bounds and shows that the Poincaré bound is superior to the Cheeger bound for simple random walk on a tree and random walk on a finite group with any symmetric generating set. This partially resolves a question posed by Diaconis and Stroock.
Citation
Jason Fulman. Elizabeth L. Wilmer. "Comparing eigenvalue bounds for Markov chains: when does Poincaré beat Cheeger?." Ann. Appl. Probab. 9 (1) 1 - 13, February 1999. https://doi.org/10.1214/aoap/1029962594
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