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

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.