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Hollis Williams
Hollis Williams
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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 joinjoint 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})$$( M_{11}, M_{22})$ as $n\to \infty$?

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 join 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 $n\to \infty$?

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 $n\to \infty$?

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Raghav
Raghav
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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 join 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 $n\to \infty$?