Expectation value of random GUE matrix

Question

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 $A.$

Answer 1

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

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