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Emergence of Giant Cycles and Slowdown Transition in Random Transpositions and k-Cycles


 
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1. Title Title of document Emergence of Giant Cycles and Slowdown Transition in Random Transpositions and k-Cycles
 
2. Creator Author's name, affiliation, country Nathanael Berestycki; University of Cambridge; United Kingdom
 
3. Subject Discipline(s)
 
3. Subject Keyword(s) Random permutations
 
3. Subject Subject classification 60J10; 60K35; 60B15; 05C80; 05C12; 05C65
 
4. Description Abstract Consider the random walk on the permutation group obtained when the step distribution is uniform on a given conjugacy class. It is shown that there is a critical time at which two phase transitions occur simultaneously. On the one hand, the random walk slows down abruptly: the acceleration (i.e., the second time derivative of the distance) drops from $0$ to $-\infty$ at this time as $n\to\infty$. On the other hand, the largest cycle size changes from microscopic to giant. The proof of this last result is considerably simpler and holds more generally than in a previous result of Oded Schramm for random transpositions. It turns out that in the case of random $k$-cycles, this critical time is proportional to $1/[k(k-1)]$, whereas the mixing time is known to be proportional to $1/k$.
 
5. Publisher Organizing agency, location
 
6. Contributor Sponsor(s) EPSRC
 
7. Date (YYYY-MM-DD) 2011-01-12
 
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/850
 
10. Identifier Digital Object Identifier 10.1214/EJP.v16-850
 
11. Source Journal/conference title; vol., no. (year) Electronic Journal of Probability; Vol 16
 
12. Language English=en
 
14. Coverage Geo-spatial location, chronological period, research sample (gender, age, etc.)
 
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