The Distribution of Permutation Matrix Entries Under Randomized Basis

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 $sY + Z$ where $Y$ is Poisson distributed, $Z$ is normally distributed, and $s$ is a constant.