Abstract
Let Y = (y 1, y 2,…), y 1 ≥ y 2 ≥ …, be the list of sizes of the cycles in the composition of cn transpositions on the set {1,2,…, n}. We prove that if c > 1/2 is constant and n → ∞, the distribution of f(c)Y/n converges to PD(1), the Poisson–Dirichlet distribution with parameter 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.
In loving memory of my parents, Hanna and Mickey Schramm
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References
O. Angel, Random infinite permutations and the cyclic time random walk, in Random Walks and Discrete Potential Theory (C. Banderier and C. Krattenthaler, eds.), Discrete Mathematics and Theoretical Computer Science, 2003, pp. 9–16, http://dmtcs.loria.fr/proceedings/html/dmAC0101.abs.html.
N. Alon and J. H. Spencer, The Probabilistic Method, second edition, Wiley-Interscience Series in Discrete Mathematics and Optimization, Wiley-Interscience, New York, 2000. With an appendix on the life and work of Paul Erdös.
N. Berestycki and R. Durrett, A phase transition in the random transposition random walk, arXiv:math.PR/0403259.
P. Diaconis, M. McGrath and J. Pitman, Riffle shuffles, cycles, and descents, Combinatorica 15 (1995), 11–29.
P. Diaconis, E. Mayer-Wolf, O. Zeitouni and M. P. Zerner, The Poisson–Dirichlet law is the unique invariant distribution for uniform split-merge transformations, The Annals of Probability 32 (2004), 915–938.
P. Diaconis and M. Shahshahani, Generating a random permutation with random transpositions, Zeitschrift für Wahrscheinlichkeitstheorie und Verwandte Gebiete 57 (1981), 159–179.
R. M. Dudley, Real Analysis and Probability, Wadsworth & Brooks/Cole Advanced Books & Software, Pacific Grove, CA, 1989.
L. Holst, The Poisson–Dirichlet distribution and its relatives revisited, 2001, http://www.math.kth.se/matstat/fofu/reports/PoiDir.pdf, preprint.
S. Janson, T. Luczak and A. Rucinski, Random Graphs, Wiley-Interscience Series in Discrete Mathematics and Optimization, Wiley-Interscience, New York, 2000.
J. Spencer, Ten lectures on the probabilistic method, Volume 64 of CBMS-NSF Regional Conference Series in Applied Mathematics, second edition, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 1994.
B. Tóth, Improved lower bound on the thermodynamic pressure of the spin 1/2 Heisenberg ferromagnet, Letters in Mathematical Physics 28 (1993), 75–84.
G. A. Watterson, The stationary distribution of the infinitely-many neutral alleles diffusion model, Journal of Applied Probability 13 (1976), 639–651.
Acknowledgement
We have had the pleasure to benefit from conversations with Rick Durrett, Michael Larsen, Russ Lyons, David Wilson and Ofer Zeitouni in connection with this work.
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Schramm, O. (2011). Compositions of Random Transpositions. In: Benjamini, I., Häggström, O. (eds) Selected Works of Oded Schramm. Selected Works in Probability and Statistics. Springer, New York, NY. https://doi.org/10.1007/978-1-4419-9675-6_17
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