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 $[-1,1]$. 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.

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.

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 $[0,1]$. 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.

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 $[-1,1]$. 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.