Spectral density of symmetrized Haar matrix

Question

Let $O$ be a random orthogonal matrix (according to Haar measure) of size $n$. I found by simulations that the spectral density of $O+O^\top$ is the arcsin law rescaled to the interval $[-2,2]$. I can't find this result in some textbooks on random matrix theory and free probability that I have. Searching on internet also give no relevant result. So I want to know about the publications on this model and in particular its rank-1 perturbation $O+O^\top + \lambda u u^\top$ where $u$ is a random unit vector.

Answer 1

Since $O$ is orthogonal, $O^\top=O^{-1}$ commutes with $O$, hence the eigenvalues $\mu_n$ of $O+O^\top$ are related to the eigenvalues $e^{i\phi_n}$ of $O$ by $\mu_n=2\cos\phi_n$. The spectral density of $O+O^\top$ is therefore simply obtained from the known spectral density of $O$ (Jacobi distribution) by a change of variables.

For large $n$ the phases $\phi_n$ are uniformly distributed in $(0,2\pi)$, and then $$P(\mu)\,d\mu=\frac{1}{\pi}|d\mu/d\phi|^{-1}\,d\mu=\frac{d\mu}{\pi\sqrt{4-\mu^2}}=\frac{1}{\pi}d\arcsin(\mu/2),\;\;|\mu|<2.$$

For exact results on rank-one perturbations, see Forrester.