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A lower bound for the mixing time of the random-to-random Insertions shuffle


 
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1. Title Title of document A lower bound for the mixing time of the random-to-random Insertions shuffle
 
2. Creator Author's name, affiliation, country Eliran Subag; Technion - Israel Institute of Technology; Israel
 
3. Subject Discipline(s)
 
3. Subject Keyword(s) Mixing-time, card shuffling, random insertions, cutoff phenomenon.
 
3. Subject Subject classification 60J10
 
4. Description Abstract

The best known lower and upper bounds on the mixing time for the random to-random insertions shuffle are $(1/2-o(1))n\log n$ and $(2+o(1))n\log n$. A long standing open problem is to prove that the mixing time exhibits a cutoff. In particular, Diaconis conjectured that the cutoff occurs at $3/4n\log n$. Our main result is a lower bound of $t_n = (3/4-o(1))n\log n$, corresponding to this conjecture.

Our method is based on analysis of the positions of cards yet-to-be removed.We show that for large $n$ and $t_n$ as above, there exists $f(n)=\Theta(\sqrt{n\log n})$ such that,with high probability, under both the measure induced by the shuffle and the stationary measure, the number of cards within a certain distance from their initial positionis $f(n)$ plus a lower order term. However, under the induced measure, this lower order term is strongly influenced bythe number of cards yet-to-be-removed, and is of higher order than forthe stationary measure.
 
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7. Date (YYYY-MM-DD) 2013-02-03
 
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/1950
 
10. Identifier Digital Object Identifier 10.1214/EJP.v18-1950
 
11. Source Journal/conference title; vol., no. (year) Electronic Journal of Probability; Vol 18
 
12. Language English=en en
 
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