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?



Largest eigenvalues of a (random) correlation matrix?
It sounds like you are asking for the distribution of the eigenvalues of a random symmetric matrix with i.i.d. centered entries. This is a huge subject, and you should read the oeuvre of Mehta, or Terry Tao's blog posts, or Anderson/Guionnet/Zeitouni, or..


*Mehta, Madan Lal*, Random matrices., Pure and Applied Mathematics (Amsterdam) 142. Amsterdam: Elsevier (ISBN 0-12-088409-7/hbk). xviii, 688 p. (2004). [ZBL1107.15019](https://zbmath.org/?q=an:1107.15019).


*Anderson, Greg W.; Guionnet, Alice; Zeitouni, Ofer*, An introduction to random matrices, Cambridge Studies in Advanced Mathematics 118. Cambridge: Cambridge University Press (ISBN 978-0-521-19452-5/hbk). xiv, 492 p. (2010). [ZBL1184.15023](https://zbmath.org/?q=an:1184.15023).


**ADDENDUM** For random *correlation* matrices, it is a little less clear how you produce them, and the results depend on the model. A detailed discussion is given in:


*Holmes, R. B.*, [**On random correlation matrices**](http://dx.doi.org/10.1137/0612019), SIAM J. Matrix Anal. Appl. 12, No. 2, 239-272 (1991). [ZBL0723.15020](https://zbmath.org/?q=an:0723.15020).


(see section 3.2 and the sequel).


