**Case 3:**

Let me define $t=x\sqrt{2\gamma}$, then it is known from random-matrix theory (see, for example, <A HREF="https://press.princeton.edu/books/hardcover/9780691128290/log-gases-and-random-matrices-lms-34">Forrester's book</A>) that for a fixed $\gamma$ the probability distribution $P(x_1)$ of a single eigenvalue $x_1$ tends in the limit $n\rightarrow\infty$ to the $\gamma$-independent semicircle
$$P(x)=\frac{1}{\pi n}\sqrt{2n-x^2},\;\;|x|\leq\sqrt{2n}.$$
The desired ratio $\nu$ then evaluates to
$$\nu=\frac{\int (2\gamma x^2-1)P(x)\,dx}{\left[\int (2\gamma x^2-1)^2P(x)\,dx\right]^{1/2}}=\frac{\gamma n-1}{\sqrt{2 \gamma n (\gamma n-1)+1}}\rightarrow \frac{1}{\sqrt 2}\;\;\text{for}\;\;n\rightarrow\infty.$$

**Case 2:**

The case that $n\rightarrow\infty$, $\gamma\rightarrow 0$ at fixed $\gamma n=\alpha>0$ has been studied in <A HREF="https://projecteuclid.org/euclid.ecp/1465320995">The mean spectral measures of random Jacobi matrices related to Gaussian beta ensembles</A> (2014), see also <A HREF="https://arxiv.org/abs/1611.09476">arXiv:1611.09476</A>. The probability distribution $P_\alpha(t)$ is given in this limit by
$$P_\alpha(t)=\frac{e^{-t^2/2}}{\alpha\sqrt{2\pi}}\frac{\Gamma(\alpha)}
{|f(t)|^2},\;\;f(t)=\int_0^\infty x^{\alpha-1}e^{ix t-x^2/2}\,dx.$$
From this the desired $\nu$ can be readily computed, 
$$\nu_\alpha=\frac{\int (t^2-1)P_\alpha(t)\,dt}{\left[\int (t^2-1)^2P_\alpha(t)\,dt\right]^{1/2}}=\frac{\alpha}{\sqrt{\alpha (2 \alpha+3)+2}},$$
so for $\alpha=1$ I find $\nu_1=1/\sqrt 7$.

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Notice the similarity between the two large-$n$ formulas for $\nu$ at fixed $\gamma$ (case 1) and at fixed $\gamma n=\alpha$ (case 2). Both give $\nu$ of order unity, but with a different value.