Pith Number

pith:H6GSNRQG

pith:2026:H6GSNRQGDM72PSQ3TRYG3CQ3NY

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Zero correlations and averaged fields of orthonormal Gaussian functions

Lu\'is Daniel Abreu, Tomoyuki Shirai

Iterated orthonormal Gaussian entire functions produce zero processes with index-dependent short-range correlations and averaged fields converging almost surely to 1.

arxiv:2605.17296 v1 · 2026-05-17 · math.PR · math-ph · math.CA · math.MP

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4 Citations open

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The bundle contains the canonical record plus signed events. A mirror can host it anywhere and recompute the same current state with the deterministic merge algorithm.

Claims

C1strongest claim

The normalized pair correlations g_{n,n+k}(z,w) exhibit repulsion for k=1, attraction for k=2, and no short-range second-order correlation for k >= 3 as w -> z; the averaged fields converge almost surely to 1 in C(K) and their scaled fluctuations converge to the Gaussian process G(z) = 1/sqrt(pi) int 1_{B(z,1)}(u) dW_R(u).

C2weakest assumption

The sequence f_n is constructed by iterated application of the Landau raising operator to the base Gaussian entire function f_0 while preserving the pointwise orthonormality condition E[e^{-|z|^2} f_n(z, bar z) bar f_{n'}(z, bar z)] = delta_{n n'}.

C3one line summary

Proves interlacing-like zero pair correlations (repulsion at k=1, attraction at k=2, none for k>=3) and a functional CLT for averaged fields of iterated Gaussian entire functions, confirming signal-processing conjectures.

References

72 extracted · 72 resolved · 1 Pith anchors

[1] L. D. Abreu,Sampling and interpolation in Bargmann-Fock spaces of polyanalytic functions. Appl. Comp. Harm. Anal., 29, 287–302, (2010) 2010

[2] L. D. Abreu, K. Gr¨ ochenig, J. L. Romero,On accumulated spectrograms. Trans. Amer. Math. Soc.368, 3629–3649, (2016) 2016

[3] L. D. Abreu, K. Gr¨ ochenig, J. L. Romero,Harmonic analysis in phase space and finite Weyl-Heisenberg ensembles. J. Stat. Phys., vol. 174, 5, 1104–1136, (2019) 2019

[4] L. D. Abreu,Local maxima of white noise spectrograms and Gaussian Entire Functions. J. Fourier Anal. Appl., vol. 28, Article number: 88 (2022) 2022

[5] L. D. Abreu, D. Alpay, T. Georgiou, P. Jorgensen,Analytic continuation of time in Brownian motion. Stochastic distributions approach. J. Math. Anal. Appl., 130438, (2026) 2026

Formal links

2 machine-checked theorem links

Receipt and verification

First computed	2026-05-20T00:03:50.783212Z
Builder	pith-number-builder-2026-05-17-v1
Signature	Pith Ed25519 (`pith-v1-2026-05`) · public key
Schema	pith-number/v1.0

Canonical hash

3f8d26c6061b3fa7ca1b9c706d8a1b6e0d7c9667d036f732e083e5c579d6d52c

Aliases

arxiv: 2605.17296 · arxiv_version: 2605.17296v1 · doi: 10.48550/arxiv.2605.17296 · pith_short_12: H6GSNRQGDM72 · pith_short_16: H6GSNRQGDM72PSQ3 · pith_short_8: H6GSNRQG

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Verify this Pith Number yourself

curl -sH 'Accept: application/ld+json' https://pith.science/pith/H6GSNRQGDM72PSQ3TRYG3CQ3NY \
  | jq -c '.canonical_record' \
  | python3 -c "import sys,json,hashlib; b=json.dumps(json.loads(sys.stdin.read()), sort_keys=True, separators=(',',':'), ensure_ascii=False).encode(); print(hashlib.sha256(b).hexdigest())"
# expect: 3f8d26c6061b3fa7ca1b9c706d8a1b6e0d7c9667d036f732e083e5c579d6d52c

Canonical record JSON

{
  "metadata": {
    "abstract_canon_sha256": "8c1c5e91750fb2c1ed472a570e338e68583ba4358a2bee7cc366669ecec0196a",
    "cross_cats_sorted": [
      "math-ph",
      "math.CA",
      "math.MP"
    ],
    "license": "http://arxiv.org/licenses/nonexclusive-distrib/1.0/",
    "primary_cat": "math.PR",
    "submitted_at": "2026-05-17T07:16:21Z",
    "title_canon_sha256": "81670575951b6bcac24707a936954d7b070d044cc4d9344560655eb5cb0733bf"
  },
  "schema_version": "1.0",
  "source": {
    "id": "2605.17296",
    "kind": "arxiv",
    "version": 1
  }
}