For past few months, I am self studying random matrices from the book - An Introduction to Random Matrices by Anderson et.al.
I was going through the result concerning the eigenvalues of the GUE at the edge of the spectrum (Theorem 3.1.4)
I am trying to understand how the non intuitive scale factor N^(2/3) comes up. The proof of Lemma 3.7.2 seems to play the key role in the proof of the above theorem.
Define $\displaystyle \psi_n(x)=\frac{e^{-x^2/4}H_n(x)}{\sqrt{\sqrt{2\pi}n!}}$ where $H_n(x)$ is the n-th Hermite polynomial and let $\displaystyle \Psi_n(x)=n^{1/12}\psi_n(2\sqrt{n}+\frac{x}{n^{1/6}})$. For this $\Psi_n(x)$ we have
I went through the proof of lemma 3.7.2 which uses steepest descent. Why that scaling factor is used is hidden in that proof. The following are first few steps of the proof.
Focus on the last paragraph of the above image.
A taylor expansion of $F(s) = log(1+s) + s^2/2 - s$ starts with $s^3/3$ in a neighbourhood of $0$, explains the particular scaling for u.
This line, present in the proof, seems to answer my question. However, I fail to understand how that explains the particular scaling of u. I am not so familiar with complex analysis. I would be highly obliged if anyone can explain the scaling ($n^{1/6}$) and the reasoning mentioned here.