Let $A$ be a matrix of the Gaussian unitary ensemble (GUE) and $v\_1,v\_2$ be two orthonormal vectors.


I wonder if one can compute (or at least get a non-trivial lower bound on) the expectation value


$$\mathbf E\sqrt{4\langle Av\_1,v\_2 \rangle^2 + (\langle Av\_1,v\_1 \rangle-\langle Av\_2,v\_2 \rangle)^2}.$$


We can choose a unitary $U$ such that $Ue\_1=v\_1$ and $U e\_2 = v\_2$ then we have


$$\mathbf E\sqrt{4\langle U^\*AUe\_1,e\_2 \rangle^2 + (\langle U^\*AUe\_1,e\_1 \rangle-\langle U^\*AUe\_2,e\_2 \rangle)^2}.$$


Since the GUE is invariant under unitaries, this implies we have to compute (estimate from below)


$$\mathbf E\sqrt{4a\_{12}^2 + (a\_{11}-a\_{22})^2},$$


where $a\_{ij}$ are the matrix entries of $A.$



Expectation value of random GUE matrix
From the definion of the GUE for $n\times n$ Hermitian matrices $A$, one has the integral expression.
$$\mathbb{E}[f({\rm Re}\,a\_{12},{\rm Im}\,a\_{12},a\_{11},a\_{22})]=\frac{n^2}{2\pi^2}\iint\!\!\!\!\!\iint\_{-\infty}^{\infty} f(x\_1,x\_2,y\_1,y\_2)$$
$$\qquad\qquad\times\exp\left(-n(x\_1^2+x\_2^2+y\_1^2/2+y\_2^2/2)\right)\,dx\_1dx\_2dy\_1dy\_2.$$


The expression in the OP is cumbersome to evaluate, a simpler example is


$$\mathbb{E}[4|a\_{12}|^2+(a\_{11}-a\_{22})^2]=6/n.$$


A numerical evaluation gives


$$\mathbb{E}\left[\sqrt{4|a\_{12}|^2+(a\_{11}-a\_{22})^2}\right]=\frac{2.25676}{\sqrt{n}}.$$


