Geodesic
Asymptotics of the Uniform Measures on Simplices and Random Compositions and Partitions
Funkcionalʹnyj analiz i ego priloženiâ, Tome 37 (2003) no. 4, pp. 39-48

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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.
Keywords: limit shape, uniform measure on a simplex.
Mots-clés : composition, partition 
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A. M. Vershik; Yu. V. Yakubovich. Asymptotics of the Uniform Measures on Simplices and Random Compositions and Partitions. Funkcionalʹnyj analiz i ego priloženiâ, Tome 37 (2003) no. 4, pp. 39-48. http://geodesic.mathdoc.fr/item/FAA_2003_37_4_a8/