We study the distribution of entries of a random permutation matrix under a “randomized basis,” i.e., we conjugate the random permutation matrix by an independent random orthogonal matrix drawn from Haar measure. It is shown that under certain conditions, the linear combination of entries of a random permutation matrix under a “randomized basis” converges to a sum of independent variables $$sY+Z$$sY+Z where Y is Poisson distributed, Z is normally distributed, and s is a constant.

中文翻译：

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随机化基础下置换矩阵项的分布

我们在“随机化基础”下研究随机置换矩阵的条目分布，即，我们通过从 Haar 测度得出的独立随机正交矩阵来共轭随机置换矩阵。结果表明，在一定条件下，“随机化基”下的随机置换矩阵项的线性组合收敛于自变量之和$$sY+Z$$sY+Z 其中Y是泊松分布，Z是正态分布分布，且 s 是常数。