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The Cutoff Phenomenon for Ergodic Markov Processes


 
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1. Title Title of document The Cutoff Phenomenon for Ergodic Markov Processes
 
2. Creator Author's name, affiliation, country Guan-Yu Chen; Department of Applied Mathematics, National Chiao Tung University
 
2. Creator Author's name, affiliation, country Laurent Saloff-Coste; Department of Mathematics, Cornell University
 
3. Subject Discipline(s)
 
3. Subject Keyword(s) cutoff phenomenon, ergodic Markov semigroups
 
3. Subject Subject classification 60J05, 60J25
 
4. Description Abstract We consider the cutoff phenomenon in the context of families of ergodic Markov transition functions. This includes classical examples such as families of ergodic finite Markov chains and Brownian motion on families of compact Riemannian manifolds. We give criteria for the existence of a cutoff when convergence is measured in $L^p$-norm, $1 < p < \infty$. This allows us to prove the existence of a cutoff in cases where the cutoff time is not explicitly known. In the reversible case, for $1 < p\le \infty$, we show that a necessary and sufficient condition for the existence of a max-$L^p$ cutoff is that the product of the spectral gap by the max-$L^p$ mixing time tends to infinity. This type of condition was suggested by Yuval Peres. Illustrative examples are discussed.
 
5. Publisher Organizing agency, location
 
6. Contributor Sponsor(s) DMS 0603886, NSF grant DMS 0306194 and NCTS, Taiwan
 
7. Date (YYYY-MM-DD) 2008-01-20
 
8. Type Status & genre Peer-reviewed Article
 
8. Type Type
 
9. Format File format PDF
 
10. Identifier Uniform Resource Identifier http://ejp.ejpecp.org/article/view/474
 
10. Identifier Digital Object Identifier 10.1214/EJP.v13-474
 
11. Source Journal/conference title; vol., no. (year) Electronic Journal of Probability; Vol 13
 
12. Language English=en
 
14. Coverage Geo-spatial location, chronological period, research sample (gender, age, etc.)
 
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