Compositions of random transpositions

By Oded Schramm. 2004

92 citations recorded by the source corpus.

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.

Subfields: Artificial Intelligence, Mathematical Physics, Statistics and Probability

View Paper: https://www.semanticscholar.org/paper/0a45fc12409ea1df167511f70d193f4f7eeda080

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Citations in the corpus

On the Cycle Structure of Mallows Permutations (2016, methodology): “Schramm [24] considered the above card shuffling scheme further, investigating the state of the deck after tn steps are performed.”
Emergence of Giant Cycles and Slowdown Transition in Random Transpositions and k-Cycles (2010, background): “(The Poisson-Dirichlet distribution describes the limiting cycle distribution of a uniform random permutation, see [19].”
Mixing times for random k-cycles and coalescence-fragmentation chains (2010, methodology): “By Nathanaël Berestycki1, Oded Schramm and Ofer Zeitouni2”