Isolated eigenvalues of a random matrix

Question

This is a continuation of this question. Let $O$ be a random orthogonal matrix (according to Haar measure) of size $n$. I want to study the eigenvalues of the matrix $O+O^\top + \lambda uu^\top$ where $\lambda>0$ and $u$ is an arbitrary unit vector in $\mathbb R^n$. I found that as $n$ goes to infinity, the spectrum consists of eigenvalues with limiting density given by the arcsin law rescaled to the interval $[-2,2]$ and an isolated eigenvalue on the right. Let $\lambda_{max}$ and $u_{max}$ be the maximum eigenvalue and the corresponding unit eigenvector. Then $$ \lim_{n \rightarrow \infty} \lambda_{max} = \sqrt{\lambda^2+4}, $$ $$ \lim_{n \rightarrow \infty} (u_{max}^\top u)^2 = \frac{\lambda}{\sqrt{\lambda^2+4}}. $$ I obtained these results by the non-rigorous replica method and verified them by simulations.

So my question is whether a rigorous proof of these results can be found in the literature (I can't find it in the reference given in the answer of the mentioned mathoverflow question). If you have a simple proof I would be eager to know.

Answer 1

I use results from Forrester's article (written for a rank-one perturbation of a matrix from the GOE, but more generally valid for an isotropic random-matrix ensemble).

A. Largest eigenvalue

Equation (2.6) in that paper gives the largest eigenvalue $\lambda_{\text max}$ of $O+O^\top+\lambda uu^\top$, with $n\gg 1$, as the solution of $$1-\lambda\int \frac{P(\mu)}{\lambda_{\text{max}}-\mu}\,d\mu=0,$$ with $$P(\mu)=\frac{1}{\pi\sqrt{4-\mu^2}},\;\;|\mu|<2,$$ the large-$n$ eigenvalue density of $O+O^\top$ calculated in my previous answer. This gives the equation $$\frac{2}{\pi \sqrt{\lambda_{\text{max}}^2-4}} \left[\arctan\left(\frac{\lambda_{\text{max}}-2}{\sqrt{\lambda_{\text{max}}^2-4}}\right)+\arctan\left(\frac{\lambda_{\text{max}}+2}{\sqrt{\lambda_{\text{max}}^2-4}}\right)\right]=\frac{1}{\lambda},$$ with solution $$\lambda_{\text{max}}=\sqrt{\lambda^2+4}.$$

B. Eigenvector of largest eigenvalue

From page 20 of Forrester I find, again for $n\gg 1$, $$(u_{\text{max}}^\top u)^2=\left(\int \frac{P(\mu)}{(\lambda_{\rm max}-\mu)^2}\,d\mu\right)^{-1}\left(\int \frac{P(\mu)}{\lambda_{\rm max}-\mu}\,d\mu\right)^2$$ $$\qquad = \frac{\lambda}{\sqrt{\lambda^2+4}}.$$