Compositions of random transpositions

LetY=(y1,y2, ...),y1≥y2≥..., be the list of sizes of the cycles in the composition ofcn transpositions on the set {1, 2, ...,n}. We prove that ifc>1/2 is constant andn → ∞, the distribution off(c)Y/n converges toPD(1), the Poisson-Dirichlet distribution with parameter 1, where the functionf is kn...

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Bibliographic Details
Published in:Israel journal of mathematics 2005-01, Vol.147 (1), p.221-243
Main Author: Schramm, Oded
Format: Article
Language:English
Subjects:
Coagulation
Composition
Dirichlet problem
Mathematics
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Summary:LetY=(y1,y2, ...),y1≥y2≥..., be the list of sizes of the cycles in the composition ofcn transpositions on the set {1, 2, ...,n}. We prove that ifc>1/2 is constant andn → ∞, the distribution off(c)Y/n converges toPD(1), the Poisson-Dirichlet distribution with parameter 1, where the functionf is known explicitly. A new proof is presented of the theorem by Diaconis, Mayer-Wolf, Zeitouni and Zerner stating that thePD(1) measure is the unique invariant measure for the uniform coagulation-fragmentation process.
ISSN:0021-2172
1565-8511
DOI:10.1007/BF02785366