Elementary bounds on Poincar e and log-Sobolev constants for decomposable Markov chains

We consider nite-state Markov chains that can be naturally decomposed into smaller \projection" and \restriction" chains. Possibly this decomposition will be inductive, in that the restriction chains will be smaller copies of the initial chain. We provide expressions for Poincar e (respectively, log-Sobolev) constants of the initial Markov chain in terms of Poincar e (respectively, log-Sobolev) constants of the projection and restriction chains, together with further parameter. In the case of the Poincar e constant, our bound is always at least as good as existing ones, and, depending on the value of the extra parameter, may be much better. There appears to be no previously published decomposition result for the log-Sobolev constant. Our proofs are elementary and self-contained. This work was done while the authors were visiting the Isaac Newton Institute for Mathematical Sciences, Cambridge, UK. y Partially supported by EPSRC grant \Sharper Analysis of Randomized Algorithms: a Computational Approach" and the IST Programme of the EU under contract IST-1999-14036 (RAND-APX). Second author also supported by an award from DAAD. Postal address: School of Informatics, University of Edinburgh, The King's Buildings, Edinburgh EH9 3JZ, United Kingdom. z Research supported in part by NSF grant DMS-0100289. Postal address: School of Mathematics, Georgia Institute of Technology Atlanta, GA 30332-0160, USA. x Supported in part by a NSF CAREER grant. Postal address: Department of Computer Science, University of Chicago, 1100 E 58th Street, Chicago, IL 60637, USA