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Given a $n\times n$ symmetric random matrix such that

  1. all diagonal elements are all fixed as $0$.
  2. all other elements in the upper triangle are uniform random variables over $[0,1]$. all values in the lower triangle are set accordingly to ensure symmetry.

The question is,

Is there known exact/or asymptotic ($n \to \infty$) formula for distribution of the largest eigenvalue of such matrix?

*Now as suggested by the comment, the exact formula seems not exist, but still hope someone could help with the asymptotic case.

I am not familiar with random matrix theory, but I think this kind of matrix is not uncommon, and there hopefully should have been some result. Thanks!

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  • $\begingroup$ an exact result for any $n$ is unlikely, but for large $n$ you should recover the GOE distribution (Gaussian distribution or uniform distribution should not make difference for large $n$). $\endgroup$ – Carlo Beenakker Jan 13 '18 at 20:05
  • $\begingroup$ Actually, the mean is $1/2$ and the variance is $1/12$, so you are in the regime that the rank 1 perturbation dominates and the fluctuations are Gaussian. Read the answer (and references) to your previous question mathoverflow.net/questions/290582/… $\endgroup$ – ofer zeitouni Jan 13 '18 at 21:46

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