The Distribution of Permutation Matrix Entries Under Randomized Basis

TL;DR

This paper investigates how the entries of a random permutation matrix behave when conjugated by a Haar-distributed orthogonal matrix, revealing a convergence to a sum of independent Poisson and normal variables under certain conditions.

Contribution

It introduces a novel analysis of permutation matrix entries under a randomized basis, showing their distribution converges to a sum of independent Poisson and normal variables.

Findings

Entries converge to a sum of independent Poisson and normal variables.

Distributional convergence under certain conditions.

Provides insights into the spectral properties of permutation matrices.

Abstract

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.