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Compositions of random transpositions

Oded Schramm
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Abstract

Let $Y=(y_1,y_2,...)$, $y_1\ge y_2\ge...$, be the list of sizes of the cycles in the composition of $c n$ transpositions on the set $\{1,2,...,n\}$. We prove that if $c>1/2$ is constant and $n\to\infty$, the distribution of $f(c)Y/n$ converges to PD(1), the Poisson-Dirichlet distribution with paramenter 1, where the function $f$ is known explicitly. A new proof is presented of the theorem by Diaconis, Mayer-Wolf, Zeitouni and Zerner stating that the PD(1) measure is the unique invariant measure for the uniform coagulation-fragmentation process.

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Oded Schramm. 2007-07-04. Compositions of random transpositions. https://arxiv.org/abs/math/0404356

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