Consider a $n\times n$ GOE random matrix. If we assume that $|\lambda\_1|>|\lambda\_2|\ge \dots \ge |\lambda\_n|$, can we get the order of $｜\lambda\_1｜/｜\lambda\_2｜$ or even $\lambda\_1/\lambda\_2$?


Any reference are appreciate!




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If we change the condition that $\lambda\_1>\lambda\_2\ge \dots \ge \lambda\_n$, then by $\lambda\_1-\lambda\_2=O\_p(n^{2/3})$ (this result follows from the limiting joint distribution of spectral edge is Tracy-Windom law), can we have that
$$
\frac{\lambda\_1}{\lambda\_2}=1+O\_P(n^{2/3})?
$$



The ratio of spectral edge of the GOE matrix
$|\lambda\_1|=2+\delta\_1$, $|\lambda\_2|=2+\delta\_2$, with $\delta\_1$ and $\delta\_2$ both of order $n^{-2/3}$, so
$$|\lambda\_1|/|\lambda\_2|=1+(\delta\_1-\delta\_2)/2+{\cal O}(n^{-4/3})=1+{\cal O}(n^{-2/3}).$$


Since you do not know the sign of $\lambda\_1$ and $\lambda\_2$, the ratio $\lambda\_1/\lambda\_2$ without the absolute value signs is undetermined: it could be close to $+1$ or close to $-1$.


