I am recently studying on eigenvalues of a (random) correltion matrix. For a $N\times N$ correlation matrix (with a given meaning of randomness), its (1st, 2nd, etc.) eigenvalues have some distributions depending on $N$. For the case $N=2$, a correlation matrix is like $$\begin{bmatrix}1&x\\x&1\end{bmatrix}$$ with $x\in[-1,1]$ and its eigenvalues are $1\pm x$ and so its largest eigenvalue has an upper bound $2$. Also one can see that the distribution of the largest eigenvalue is actually a uniform distribution on $[1,2]$, when this correlation matrix is sampled uniformly ($x$ uniform in $[-1,1]$).

One would be interested to know the analogue in general dimension, or even the limit $N\rightarrow\infty$, i.e. distribution or bounds for the largest eigenvalue. I looked through literatures and didn't find any exact answers. Is there any development on this topic? Or is it still an open problem?