Asymptotics of the Uniform Measures on Simplices and Random Compositions and Partitions
Asymptotics of the Uniform Measures on Simplices and Random Compositions and Partitions
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单纯形和随机组合与划分的统一测度的渐近性
DOI：
10.1023/b:faia.0000015578.02338.0e
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发表时间：
2003
影响因子：
0.4
通讯作者：
Y. Yakubovich
中科院分区：
文献类型：
--
作者：
A. Vershik;Y. Yakubovich
我们研究当维数趋于无穷大时，有限维单纯形上均匀测度的极限行为，以及该问题的离散模拟，即组合上均匀测度的极限行为。结果表明，单纯形中典型点的坐标分布以及给定被加数数量的典型组合中被加数的分布是指数分布。我们应用这些断言来获得对具有给定被加数数量的分区的极限形状的结果的更透明的证明，细化与 Erdős 和 Lehner 关于渐近不存在重复被加数的定理相关的分区中被加数数量的估计，并概述了该估计的锐度的证明。
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.