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We say that an $n\times n$ matrix $A_n$ is a random symmetric Bernoulli matrix if each entry $a_{ij}$ is $\pm 1$ with probability $1/2$, the entries $a_{ij}$ with $i\ge j$ are independent and $a_{ij}=a_{ji}$. I would like a reference (and ideally an elementary proof) of the following fact: $$\text{var}(\lambda_1(A_n))\sim n^{-1/3}$$

I would also be happy with an upper bound $\text{var}(\lambda_1(A_n))\le f(n)$ with $\displaystyle\lim_{n\rightarrow \infty} f(n)=0$ if the proof is simple enough.

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    $\begingroup$ "of the following fact" -- Why is this a fact? $\endgroup$ Commented Feb 20 at 18:52
  • $\begingroup$ See remark 2.6 on page 18 of van Handel's lecture notes: web.math.princeton.edu/~rvan/APC550.pdf $\endgroup$ Commented Feb 20 at 19:07
  • $\begingroup$ I see no proof or reference there. $\endgroup$ Commented Feb 20 at 19:36
  • $\begingroup$ The paper [arxiv.org/pdf/2207.00546] shows that the fluctuations of the largest eigenvalue of $H := \frac{1}{\sqrt{n}}A_n$ around $2$ are of order $O(n^{-2/3})$, so the variance of order $O(n^{2(-2/3+1/2)} = n^{-1/3})$ is believable. $\endgroup$ Commented Feb 20 at 19:48

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I believe this result follows from Corollary 15 in the paper of Tao and Vu, Random Matrices: sharp concentration of eigenvalues, Random Matrices Theory Appl. 2 (2013), no. 3, 1350007, 31 pp.

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