Joint distribution of minor of Wigner Hermitian matrices Let $A$ be an $n\times n$ random matrix with i.i.d entries (say standard Gaussian) $A_{ij}$. I know that there is a CLT type result known for the determinant of $A$. More precisely there is a CLT for $\log(|\det(A)|)$. For any fixed $(i, j)$, If we look at the minor $M_{ij}$ of $A$, it follows that $\log(|M_{ij}|)$ also satisfies the same CLT.
I want to know if anything is known about the joint distribution of $(M_{11}, M_{22})$? I would be more generally interested in understanding the joint law of $(M_{ij}: 1\leq i, j\leq k)$ for an arbitrary (but fixed) $k$. In particular, can one expect any kind of asymptotic independence between $( M_{11}, M_{22})$ as $n\to \infty$?
 A: There is certainly no asymptotic independence between $\det M_{11}, \det M_{22}$.  From the base times height formula for parallelepipeds we see that
\begin{align*} \frac{|\det M_{12}|}{|\det M_{22}|} &= \frac{|v_1 \wedge v_3 \wedge \dots \wedge v_n|}{|v_2 \wedge v_3 \wedge \dots \wedge v_n|}\\
&= \frac{\mathrm{dist}(v_1, V)}{\mathrm{dist}(v_2,V)}\\
&= \frac{|v_1 \cdot u|}{|v_2 \cdot u|}
\end{align*}
where $v_i = (A_{i1}, A_{i3}, \dots, A_{in})$, $V \subset {\bf R}^{n-1}$ is the span of $v_3,\dots,v_n$, and $u$ is a unit normal to $V$.  For fixed $V$, $v_1 \cdot u, v_2 \cdot u$ are independent gaussians, and so we conclude that $\log |\det M_{12}| - \log |\det M_{22}|$ is bounded in probability, and similarly for $\log |\det M_{12}| - \log |\det M_{11}|$, hence
$\log |\det M_{11}| - \log |\det M_{22}|$ is also bounded in probability.  On the other hand, as you observed it is known that $\log |\det M_{11}|$, $\log |\det M_{22}|$ are asymptotically gaussian with variance comparable to $\log n$, so there is no asymptotic independence here.
