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 $n\to \infty$?



Joint distribution of minor of Wigner Hermitian matrices
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.


