This article is cited in 8 papers

Asymptotics of the Uniform Measures on Simplices and Random Compositions and Partitions

A. M. Vershik^ab, Yu. V. Yakubovich<sup>b

a</sup> International Erwin Schrödinger Institute for Mathematical Physics
^b St. Petersburg Department of V. A. Steklov Institute of Mathematics, Russian Academy of Sciences

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.

Keywords: limit shape, composition, partition, uniform measure on a simplex.

UDC: 519.214

Received: 15.09.2003

DOI: 10.4213/faa167

 English version: Functional Analysis and Its Applications, 2003, 37:4, 273–280

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