A constant-intensity obstruction to negatively correlated GAF zeros

Definitions and mathematical statement

A Gaussian entire function (GAF) is interpreted here as a centered circular complex Gaussian process with almost surely entire sample functions. Let $N(A)$ count zeros in a bounded Borel set. A weak negative-correlation target is $\operatorname{Cov}(N(A),N(B))\le0$ for disjoint $A,B$; a pair-intensity formulation is $\rho_2(z,w)\le\rho_1(z)\rho_1(w)$ away from the diagonal. Strong negative association would be a further requirement. The original does not fix which meaning is intended.

Original Question 5 (Balint Virag). Find a Gaussian entire function with negatively correlated zeros. The source says there is such an example on the unit disk and relates the question to repulsion of random-polynomial zeros. It does not explicitly exclude finite-rank or finite-zero examples.

Attached unreviewed literal example and obstruction. For independent standard complex Gaussians $\xi_0,\xi_1$, the aid notes \[G(z)=e^z(\xi_0+\xi_1z)\] has exactly one zero almost surely. Its zero process is a rank-one projection determinantal process, and for disjoint sets the count covariance is minus the product of the two single-zero probabilities. The aid explicitly sets this finite-rank example aside as degenerate relative to the intended infinite-zero problem.

For a nondegenerate entire GAF with constant intensity $\rho_1=L/\pi$, $L>0$, the aid asserts that its zero law equals that of the planar GAF with covariance $e^{Lz\overline w}$, up to multiplication by a zero-free entire factor. This uses the stated centered circular GAF convention and does not impose only stationarity without a Gaussian-kernel hypothesis. Its normalized pair correlation is reported strictly greater than one when $t=L|z-w|^2/2\ge2$.

More quantitatively, for \[A_L=D\left(0,\frac1{4\sqrt L}\right),\qquad B_L=D\left(\frac3{\sqrt L},\frac1{4\sqrt L}\right),\] the aid claims \[\operatorname{Cov}(N(A_L),N(B_L))\ge\frac1{128(e^{49/4}-1)}>0.\] Thus the constant-positive-intensity class cannot satisfy even the weak disjoint-count negative-correlation target. This does not rule out inhomogeneous infinite-zero entire GAFs.

The aid further discusses the standard hyperbolic disk family. In the usual expanding-disk normalization its covariance is $(1-z\overline w/R^2)^{-\alpha_R}$. Keeping the determinantal parameter $\alpha_R=1$ while $R\to\infty$ loses local zero intensity; maintaining positive limiting intensity $\alpha_R/R^2\to L>0$ instead gives the planar kernel $e^{Lz\overline w}$ and the reported obstruction. This is about the scaling limit, not a claim that the parameter-one disk GAF itself has no zeros. The aid leaves the inhomogeneous infinite-zero question open. All kernel, correlation and status assertions require independent review.

Short English statement

The literal question permits a degenerate one-zero Gaussian example. For the intended infinite-zero problem, the aid rules out constant positive intensity through an explicit positive zero-count covariance. A disk-to-plane scaling discussion does not produce a negative-correlated entire example; the inhomogeneous case remains unresolved in the report.

Sources

• Local UnsolvedMath registry, permanent ID 20000630, source code catalogue entry; aim-analysis-notes.json, notes[161], tag question.
• AIM workshop Random analytic functions, Question 5 (Balint Virag), compiled by Swaminathan Sethuraman, April 17, 2006: https://aimath.org/WWN/randomzeros/arcc1.pdf. Record status reconstructed_unverified, no clean statement. No PDF was consulted.
• Attached machine-generated research attempt 1: result partial_result, problem status at run open, rigor label proved, novelty confidence low. It is not peer reviewed and requires independent verification before citation. No authoritative bibliography for the quantitative obstruction or disk example is supplied.