Determinantal process conjecture for zeros of Gaussian matrix-valued power series

At least 19 years old · documented by 2006

Let $A_k$ , for $k\geq 0$ , be independent $n\times n$ matrices whose entries are independent standard complex Gaussian random variables. The zeros in the unit disk of the determinant of the matrix-valued Gaussian power series

A_0+zA_1+z^2A_2+\cdots

form a determinantal point process on $\mathbb D$ with kernel

\mathbb K(z,w)=\genfrac{}{}{}{}{n}{\pi}\genfrac{}{}{}{}{(1-|z|^2)^{(n-1)/2}(1-|w|^2)^{(n-1)/2}}{(1-z\overline{w})^{n+1}}.

Here the reference measure is Lebesgue measure on $\mathbb D$ . Equivalently, $\mathbb K$ is the projection kernel onto the subspace of analytic functions in $L^2\left(\mathbb D,\genfrac{}{}{}{}{n}{\pi}(1-|z|^2)^{n-1}\,dm(z)\right)$ . Determinantal process conjecture. The asserted determinantal-process description and kernel hold.

References

Primary source

Manjunath Krishnapur, “Zeros of Random Analytic Functions”, arXiv:math/0607504 (2006).

determinantal-processes random-analytic-functions random-matrices

Posted 1 month ago

Progress summary

Refreshed 1 month ago

Claimed solved

A 2007 paper claims to prove the conjecture for every matrix size, but this report does not independently verify that proof.

The conjecture says that the zeros of the determinant of a random matrix-valued power series have the stated determinantal-process law. It was previously posed in a thesis, which obtained only limited correlation information.

Known results

A thesis established only the first and second joint intensities, not the full determinantal-process description.

2007 claimed proof

Theorem 4 of a 2007 paper claims the full result for every $n \ge 1$ , using a direct construction of the determinantal process and identification with the limiting zero process of finite random matrix polynomials. Its weighted-measure kernel is equivalent to the kernel in the problem statement; the claim is reported here as unverified.

Current status (as of August 2026): A published or preprint source claims the conjecture is solved for every $n \ge 1$ , while this automated report has not independently verified the proof.

Sources

No solutions have been posted yet.