Random bistochastic matrices
Random bistochastic matrices
复制标题
DOI：
10.1088/1751-8113/42/36/365209
复制
发表时间：
2009-09-11
影响因子：
2.1
通讯作者：
Zyczkowski, Karol
中科院分区：
文献类型：
--
作者：
Cappellini, Valerio;Sommers, Hans-Juergen;Zyczkowski, Karol
研究了随机、随机和双随机矩阵的集合。虽然随机矩阵的所有列可以独立选择，但双随机矩阵的行和列必须是相关的。对给定的随机矩阵集合，应用Sinkhorn算法计算双随机矩阵的Birkhoff多面体的概率测度。对于N = 2阶矩阵，我们导出了列按狄利克雷分布的随机随机矩阵的概率分布的显式公式。对于任意N，我们构造了随机矩阵的初始集合，使我们能够根据Birkhoff多面体中心的局部平坦分布生成随机双随机矩阵。此时概率密度的值使我们能够得到Birkhoff多面体体积的估计，与最近的渐近结果一致。
Ensembles of random stochastic and bistochastic matrices are investigated. While all columns of a random stochastic matrix can be chosen independently, the rows and columns of a bistochastic matrix have to be correlated. We evaluate the probability measure induced into the Birkhoff polytope of bistochastic matrices by applying the Sinkhorn algorithm to a given ensemble of random stochastic matrices. For matrices of order N = 2 we derive explicit formulae for the probability distributions induced by random stochastic matrices with columns distributed according to the Dirichlet distribution. For arbitrary N we construct an initial ensemble of stochastic matrices which allows one to generate random bistochastic matrices according to a distribution locally flat at the center of the Birkhoff polytope. The value of the probability density at this point enables us to obtain an estimation of the volume of the Birkhoff polytope, consistent with recent asymptotic results.