Abstract
We study the limiting behavior of uniform measures on finite-dimensional simplices as the dimension tends to infinity and a discrete analog of this problem, the limiting behavior of uniform measures on compositions. It is shown that the coordinate distribution of a typical point in a simplex, as well as the distribution of summands in a typical composition with given number of summands, is exponential. We apply these assertions to obtain a more transparent proof of our result on the limit shape of partitions with given number of summands, refine the estimate on the number of summands in partitions related to a theorem by Erdős and Lehner about the asymptotic absence of repeated summands, and outline the proof of the sharpness of this estimate.
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References
G. Andrews, The Theory of Partitions, Encyclopedia of Mathematics and Its Applications, Vol. 2, Addison-Wesley, 1976.
R. Arratia, A. D. Barabour, and S. Tavaré, “Logarithmic combinatorial structures: a probabilistic approach,” in preparation.
S. Corteel, B. Pittel, C. D. Savage, and H. S. Wilf, “On the multiplicity of parts in a random partition,” Random Structures and Algorithms, 14, No. 2, 185–197 (1999).
W. Eberlein, “An integral over function space,” Canad. J. Math., 14, 370–384 (1962).
P. Erdős and J. Lehner, “The distribution of the number of summands in the partitions of a positive integer,” Duke Math. J., 8, 335–345 (1941).
H. Gupta, “On an asymptotic formula in partitions,” Proc. Indian Acad. Sci., A16, 101–102 (1942).
H.-K. Hwang and Y.-N. Yeh, “Measures of distinctness for random partitions and compositions of an integer,” Adv. Appl. Math., 19, No. 3, 378–414 (1997).
J. F. C. Kingman, “Random discrete distributions,” J. Roy. Statist. Soc. B, 37, 1–15 (1975).
A. Knopfmacher and M. E. Mays, “Compositions with m distinct parts,” Ars Combinatorica, 53, 111–128 (1999).
J. Pitman and M. Yor, “The two-parameter Poisson-Dirichlet distribution derived from a stable subordinator,” Ann. Probab., 25, 855–900 (1997).
L. Shepp and S. Lloyd, “Ordered cycle lengths in a random permutation,” Trans. Amer. Math. Soc., 121, 340–357 (1966).
M. Szalay and R. Turan, “On some problems of statistical theory of partitions. I,” Acta Math. Acad. Sci. Hungary, 29, 361–379 (1977).
H. N. V. Temperley, “Statistical mechanics and the partition of numbers. II. The form of crystal surfaces,” Proc. Cambridge Philos. Soc., 48, 683–697 (1952).
A. M. Vershik and S. V. Kerov, “Asymptotics of the maximal and typical dimension of the representations of the symmetric groups,” Funkts. Anal. Prilozhen., 19, No. 1, 25–36 (1985).
A. M. Vershik and A. A. Schmidt, “Limit measures arising in the asymptotic theory of symmetric groups, I, II,” Probab. Theory Appl., 22, No. 1, 72–88 (1977); 23, No. 2, 42–54 (1978).
A. Vershik and Yu. Yakubovich, “Limit shape and fluctuations of random partitions of naturals with fixed number of summands,” Moscow Math. J., 1, No. 3, 457–468 (2001).
A. M. Vershik, “Statistical mechanics of combinatorial partitions and their limit configurations,” Funkts. Anal. Prilozhen., 30, No. 2, 19–30 (1996).
H. Wilf, “Three problems in combinatorial asymptotics,” J. Combin. Theory, Ser. A, 35, 199–207 (1983).
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Vershik, A.M., Yakubovich, Y.V. Asymptotics of the Uniform Measures on Simplices and Random Compositions and Partitions. Functional Analysis and Its Applications 37, 273–280 (2003). https://doi.org/10.1023/B:FAIA.0000015578.02338.0e
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DOI: https://doi.org/10.1023/B:FAIA.0000015578.02338.0e