Moderate Deviations for Chaotic Analytic Zeros

Definitions and mathematical statement

Let $Z$ be the counting measure of the chaotic analytic zero process from the report, whose points are zeros of a Gaussian random entire function. For an admissible test function $h_r$, the customary centered linear statistic is \[ Z_0(h_r)=\int h_r\,{\,\mathrm{d}} Z-{\mathbb{E}}\int h_r\,{\,\mathrm{d}} Z. \] The macroscopic convention often has $h_r(z)=h(z/r)$; the record does not supply this definition, the precise Gaussian entire-function law, or the admissible class of $h$, so these remain to be confirmed rather than imposed as source facts.

The corrected question asks whether the Gaussian moderate-deviation logarithmic tail asymptotic stated in the report continues to hold in the intermediate regime \[ \frac{r}{(\log r)^2}\ll c\ll r\qquad(r\to\infty). \] With the intended asymptotic-separation reading of $\ll$, this concerns choices $c=c(r)>0$ satisfying \[ \frac{c(r)(\log r)^2}{r}\longrightarrow\infty, \qquad \frac{c(r)}r\longrightarrow0. \] The displayed lower endpoint is $r/(\log r)^2$, not $(\log r)^2$. The supplied source-verification note explicitly corrects the latter expression in the raw record.

The tail event, variance normalization, coefficient of the Gaussian exponent, test-function hypotheses, and pointwise versus uniform interpretation across $c(r)$ are not reproduced. Consequently no specific tail formula is substituted for the report's referenced asymptotic.

The raw record also asks a broader moderate-deviation and associated linear-response question for stationary random fields under hypotheses such as exponential correlation decay and finite exponential moments. The verification note identifies that as a separate source item, represented locally by OWR-1970-004, and it is not merged into the main chaotic-zero question here.

Short English statement

Does the report's Gaussian logarithmic tail asymptotic for centered chaotic-zero statistics extend to $r/(\log r)^2\ll c\ll r$? The corrected lower scale is retained; the precise statistic and tail formula need recovery.

Sources

• Local UnsolvedMath registry, permanent ID $30000964$, source code catalogue entry, "Moderate Deviations for Chaotic Analytic Zeros." No individual proposer is supplied.
• Adjacent local records OWR-1970-001 (permanent ID $30000963$) describe the Gaussian entire-function zero process, and OWR-1970-004 (permanent ID $30000965$) carries the separate stationary-field linear-response question. They do not supply this record's complete tail asymptotic.
• Stochastic Analysis (2009), Oberwolfach Reports, DOI 10.4171/owr/2008/25, pp. 1361--1424: https://doi.org/10.4171/owr/2008/25. This is the sole supplied external reference.
• Supplied literature triage, checked 2026-08-21: status "open." It reports a moderate-deviation and linear-response question under exponential correlation decay and finite exponential moments, later restricted-model results, and no located general theorem under the stated hypotheses. The verification note separates that broad question from the corrected chaotic-zero item. This dated assessment derives from supplied OWR curation; time-sensitive status and cited claims require independent verification.